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Proving 0.999... Is Equal To 1
eldavojohn writes "Some of the juiciest parts of mathematics are the really simple statements that cause one to immediately pause and exclaim 'that can't be right!' But a recent 28 page paper in The Montana Mathematics Enthusiast (PDF) spends a great deal of time fielding questions by researchers who have explored this in depth and this seemingly impossibility is further explored in a brief history by Dev Gualtieri who presents the digit manipulation proof: Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1."
(0.999...)st Post!
I was able to prove that with even one less "9" after the decimal point, it STILL equaled 1. I plan on doing this for a few more iteration until I can prove that . = 1
See my journal for slashdot ID's by year. Mine created in 2005. http://slashdot.org/journal/289875/slashdot-ids-by-year
Someone disproved math. Kids around the world celebrating. Accountants are lighting themselves on fire. Corporate greed accellerates. 'Office Space' now seen as a prophecy.
0.999... = 1 is second place to the Monty Hall Problem on the list of things that people have difficulty understanding and accepting the proof of. It is second place because the only department where I do not see graduate students giving me a confused look is the math department; with the Monty Hall problem, I will sometimes get a confused look even from people in the math department.
The other reason I put it in second place is that most people have difficult understanding the problem at all, whereas very few people have trouble understand what the Monty Hall problem is asking.
Palm trees and 8
I have wikipedia too...: "When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 9, which is 0. The final step uses algebra:"
my karma will be here long after I'm gone
Now I can replace my SLA with 100% uptime.
Wouldn't 10a (subtract) .999 be exactly 8.991...which breaks the whole "breakthrough"?
Given that 'a' is a known value of .999...
Math...it's so simple, only a mathemtician can't do it.
1/3 = 0.3333...
2/3 = 0.6666...
0.3333.... + 0.6666.... = 0.9999....
1/3 + 2/3 = 1 = 0.9999.....
.999 * 10 = 9.99
Whale
They damn well better, how else will I measure out all this dental floss.
The proof I do in my classes uses the formula for summing a geometric series.. .999.. = .9*10^0 + .9*10^-1 + .9*10^-2 + .....
= .9/(1-(1/10)) = .9/.9 = 1
a = b
a^2 = ab
a^2 - b^2 = ab - b^2
(a+b)(a-b) = b(a-b)
a + b = b
2b = b
2 = 1
Dear Slashdot: next time you want to mess with the site, add a rich-text editor for comments.
Though im sure its far from mathematically sound, Ive used this method to convince myself and others of the general "truthiness" of the .99999 = 1 debate in the past:
.3333 + .3333 + .3333 = .9999
.9999 = 1
1/3 + 1/3 + 1/3 = 1
In decimal form:
So,
the video's been on metacafe since 2007, and I'm pretty sure I learned this in school many years ago.
my karma will be here long after I'm gone
This is so old...
Even Blizzard issues a press release about it years ago because people kept arguing about it on the Blizzard forums.
http://www.mbdguild.com/index.php?topic=14915.0
You can tell how powerful someone is by the magnitude of the crime they can commit and be able to get away with.
9.999... -- 0.999... = 9 ? why ? If I suppose than 9.999... -- 0.999... limits on 9 because the operation consists of a infinite number of finite operations . In this case , 0.99999 limits on 1 , and this has sense
Humans are used to natural numbers because they're simple. But do natural numbers even exist in the real world? For the vast majority of practical purposes, 0.99999 can be thought of as one. But "one" itself is usually just a construct in the real world. There is no such thing as the perfect one of anything. The more precise we get, the more "one" becomes more of a mathematical ideal than a reality. So we spend our entire lives rounding off, because that's practical. We teach kids to count 1, 2, 3, 4... We can't very well teach them to count 0.000001, 0.00001, 0.0001, 0.001... (or any of the infinite variations of "counting" without resorting to natural numbers).
Proving that 0.99999 = 1 is an interesting intellectual exercise. But in the real world, we do it every minute of every day.
In other words--eh, close enough.
SJW: Someone who has run out of real oppression, and has to fake it.
So after all these years, has Intel been vindicated?
You are not alone. I have shown this proof to a lot of people, and I have even proved it multiple ways to those people, and I am still confronted with "this cannot be right."
Palm trees and 8
So 2+2 really is 5?
Blender And Linux Fan
In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"
The mathematician said: "Never."
The physicist said: "In an infinite amount of time."
The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."
0.99999... is equal to 1, then 0.999999...8 is equal to 0.99999... and 0.9999999...7 is equal to 0.999999...6 etc etc etc until 1 = 0! Holy shit!
Or we could just admit that using a tool incorrectly produces idiotic results.
Seven puppies were harmed during the making of this post.
That just hurt my brain and made sense at the same time...
Is it any wonder that The Big Bang Theory is one of my favourite shows?...
Soooo before my coffee, it looks like this is just them moving the problem area infinitely far away. If you just start with 0.99 and do the same thing, you can see that the numerator =/= the denominator. This is kind of like taking a derivative, throwing away the differential parts because they're "so small anyway", then reintegrating to get your answer.
*blinks* Need coffee and donuts.....
That's a real crowd-pleaser at parties. Personally I like the "writing an executable java program without a main method" trick to impress the ladies myself--that is, if I ever get to meet an actual lady who would even get that trick.
SJW: Someone who has run out of real oppression, and has to fake it.
They have mathematics in Montana?
The surprise is that someone can read wikipedia in Montana where they have had this information complete with the same proof for years.
I always used
1/3 = .33333.... .66666....
2/3 =
1/3 + 2/3 = 3/3 .333333.. + .666666666.... = .999999.....
I wish that would fit in my sig.
Meta will eat itself
16/64 cross out the 6s and you get 1/4. Or Pi is exactly 3!
But you cannot reach infinity so this is a moot point.
I think you just dismissed most of mathematics.
Actually, if you define 0.999... as having an infinite number of decimal points, then it is true. And that's how that ellipsis is defined! It means exactly infinite repeating decimals.
You've demonstrated the first hurdle that this problem raises in people's brains: they start thinking about adding "one more" decimal point to the expression, meaning they're thinking of a large but finite number of decimal points. And the second hurdle: people find it hard to believe that you can do mathematics with "infinity" as a meaningful quantity.
No kidding!!! What do you say at this point?
Oh, yeah. I guess you can tell that my math scores might have been better than my reading comprehension scores.
Disregard.
Long signatures suck.
?At first I was thinking second in difficulty, and then I read your "The other reason I put it in second place is that most people have difficult understanding the problem at all, whereas very few people have trouble understand what the Monty Hall problem is asking."
Decimal numbers are just names for points on the real number line (relative to a chosen point we call "0"). Thus one reason 0.999... is equal to 1 is that if they were referring to two different point on the number line, there would have to be a point (acutally infinitely many points) between them. Since every point on the real line can be written as a decimal (this is called the completeness property of the reals), and there is clearly no decimal greater than 0.999... and less than 1, then 0.999... and 1 must be the same point on the real line: the same number.
On a dare I proved this decades ago. Its really easy and took less than 10 minutes.
The issue is really one of notation. 1E0 also equals 1. It is not that 0.99999... is close to 1.0 It is actually equal to 1.0 and just another way to write 1.0.
Why is this is slashdot?
For instance, one could take the perspective of analysis. In the real numbers, given a number such as 1 and some other number x, if |1-x| e for any positive real number e then 1 = x (think about this for awhile, if you haven't before, and you will probably believe it). The point here is that the sequence of numbers .9, .99, .999, .9999 etc gets arbitrarily close to the number 1. So the limit of this sequence .999... = 1. I think that this proof is much more intuitive and less "tricky" (ie. does not rely on algebraic manipulation/slight of hand).
In real numbers, there is no such thing as the number 8.99999..991
There is in hyperreals, but not reals. Where are you reading that from?
Far from true. A rational number is a number you could get by expressing as a ratio (real number divided by real number). Any infinite repeating decimal is easily shown as a ratio (and often of simple integers to boot), i.e., a rational number. 0.22222... is 2/9. 0.456456456456456... is 456/999. And so on.
[
The series is infinite, you don't lose one.
Just because you can not show the number as a whole does not mean
you can not perform operations using it.
i.e. Think of pi.
You cannot reach infinity, therefore you can't reach the end of the series, therefore you can't reach the 1 at the end of the series. Which means you can't use the 1 at the end of the series to disprove the proof.
The whole point of "infinity" is that there IS no end. You can't say "but suppose there is" and use that to prove something.
I'd rather you rationally disagree than irrationally agree.
and 12+34...=1/4 Which, buy the way Euler proved by similar method http://math.ucr.edu/home/baez/qg-winter2004/zeta.pdf http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7
One time back when the Quarians still had a planet, the Reapers tricked the Geth into thinking that 1.3382 was really worth 1.3381. Hilarity ensued.
7/7 = 1.
So, why can't 3/3 = 1?
Next!
Athy, athier, athiest.
First off, by multiplying by ten, they lost one 9 at the end of the series.
I am not able rightly to apprehend the kind of confusion of ideas that could provoke such an objection. Much less why such incoherence would be modded informative.
First of all, yes, this is an infinitely repeating decimal and second, no it is not approaching 1, it is equal to 1. The proof is correct as are the others (1/9 = 0.111111..., multiply both sides by 9, simplify fraction on the left, 1 = 0.9999999..., (notice the =, no approaching done here) the end). Infinity has nothing to do in this argument (just because a number is infinitely long doesn't make anything about it moot - how would we deal with sqrt(2), pi, e, etc otherwise?)
You clearly have no clue about mathematics and misunderstand infinity. That's fine though. Life is a learning process and hopefully you learn something now. But the idiots who modded you informative should really have known better. Factually wrong statements are not informative and if you can't tell right from wrong, you shouldn't mod.
Decimal fractions are just a representation we humans use. The fundamental property of the real numbers is that it's the complete ordered field. The rest is just how we put those numbers on paper.
________
Entranced by anime since late summer 2001 and loving it ^_^
The number of things wrong with this statement are baffling.
Infinity is not an irrational number. It's not actually a number at all.
0.999... is not an irrational number. It's a rational number, as it can be expressed as a fraction. Irrational numbers cannot be expressed by any repeating set of decimal numbers.
You cannot "set" x to infinity. You cannot multiply infinity by 2. That's like trying to multiply the color red by 2. It simply isn't meaningful. Comparing the sizes of two infinite sets is a very different operation from comparing the sizes of two numbers.
The mathematics of comparing infinite sets does not in any way apply to arithmetical operations on infinitely repeating decimals.
0.999...99 is not the same as 0.999.... The former will, in fact, be less than 1, because it terminates.
People who have no concept of an infinite number of anything are SO amusing: "they lost one 9 at the end of the series"
Please tell me you were joking?
"Flame away, I wear asbestos underwear"
That would be true if there was a last nine. Since the line is infinite, there is none.
Oh... They aren't empty. The aliens live in them now. They think the high radiation is good for their complexion.
No, Ziaxia, I wasn't telling them anything on slashdot, GET OUT OF MY HEAD! GET OUT OF MY HEAD! AHHHHHH!!!! Don't make me explode! ^h^h^h^h^h^hcarrier lost
Self proclaimed typo king, and inventor of the bear destroying coffee table (patent not pending).
Therefore, Fractions are Good. Decimals are Evil!
Agreed. While I haven't seen the exact plans for 9/11 I'm pretty sure that they used decimals when calculating the fuel ...
...
Wait a minute! 9/11 = 0.81818181
Oh. My. God. Alert the truthers!
My work here is dung.
First off, by multiplying by ten, they lost one 9 at the end of the series.
Your objection assumes there is an end to the series. There isn't an end to the series, so your objection is moot.
The real question is: what do we mean by 0.999...? If it's a number with an infinite number of digits, then the arguments based on algebra are correct and 1=0.999...
First, infinite repeating decimals isn't a matter of "argument". It's the definition of 0.999... That is what the symbolism means--an infinite number of 9's.
Second, tacking a zero on to the end of a number is one method of manipulating symbols to carry out multiplication by 10. But I don't think you could say this is the "essence" of multiplication by 10. I'm not a math professor, so I can't bring a proof to the table, but I think it's reasonable to reason that multiplication by 10 can also be processed by "shifting" the number to the left one place. This is typically how computers do multiplication and division (as I understand it). If you shift an infinite number of 9's, you do not lose a 9. Infinity is infinite and it's weird. Infinity divided by ten is still infinitity.
Just because we can't reach infinity doesn't mean we can't conceptualise it.
If we can't have an infinite number of decimals than the limit must be a finite number. So what do you consider to be the limit for the number of 9's that we can add to the end?
1/9 = 0.111...
9*0.111... = 0.999 = 9* (1/9) = 9/9 = 1
so 0.999... = 1
if you treat it as a limit, it will be one. lim x-> 9 x/9 = 1
\displaystyle\lim_{x\to9}\frac{x}{9} = 1
Which is also the same as the derivative.... d/dx x/9 = 1/9, assuming the point (9,1) => y=(1/9)(x-9)+1 => y=1/9x => f(9) = 1
\frac{d}{dx} \frac{x}{9} = \frac{1}{9}
The first line should obviously be "potential vs actual infinity".
________
Entranced by anime since late summer 2001 and loving it ^_^
Huh? What sort of a proof is this First of all if a=0.999....... then 10a doesn't result in a 9.999...where the decimal precision of the two are the same. Both tend to infinity, but at some microscopic level a will always have a decimal precision different than 10a. (think of it as willie coyote always chasing road runner!) which means 10a- a ~= 9a (not exact but around 9a) and therefore 0.999.... ~= 1 i.e. 0.999.. tends to 1..No shit Sherlock!
No, you're saying .999 = .999... the elipses imply infinite repeating. Since infinity is a direction, not a value, .999... * 10 will be 9.999... there will bo no new 0 at the end of the value, ever.
Wrong, wrong and wrong.
First off, you're not talking about sets, but separate finite numbers.
Then, infinity is neither rational nor irrational.
Then, all numbers that have "infinite repeating decimals" are rational. See : http://en.wikipedia.org/wiki/Rational_number
So that means 0.999999..... is rational. Which rational you ask? Why! 9/9 :D
Finally, if you say 0.99999999..... is less than 1 : what is the difference between both?
We know it's less than any positive epsilon (0.1, 0.01, or 0.00000.....00001).
Which means it's nil.
There's no place for a single mosquito fart between 0.999999... and 1.
-2 = -2 :)
1-3 = 4-6
1-3+9/4 = 4-6+9/4
(1-3/2)^2 = (2-3/2)^2
sqrt((1-3/2)^2) = sqrt((2-3/2)^2)
1-3/2 = 2-3/2
1 = 2
No, I'm not from the US. It constantly surprises me that this mathematical curiosity takes people off guard on the net.
If this can be part of basic maths education in a country, there is no reason it couldn't be taught everywhere.
(The reason I remember this problem and when I learned about it was because when I was shown the proof for it I thought it's particularly cool and finally, something interesting came along in maths. It kindled a fondness for mathematics in me.)
It takes a man to suffer ignorance and smile
Be yourself no matter what they say
This just goes to show that people don't really know what numbers are, at least when they are infinite decimal numbers. A finite decimal number corresponds to a rational number, e.g. 9.99 corresponds to 9 + 9/10 + 9/100. The way you describe infinite decimal numbers of by denoting a sequence of finite decimal numbers that goes towards this infinite decimal, in our case: 0.9, 0.99, 0.999, etc. This, by the way, is how you construct the real numbers (pi is described in such a way).
In doing so, however, there are multiply ways of describing the same number; the sequences 0.9, 0.99, 0.999, etc. and 1, 1, 1, etc. describe the same number, and this apparent non-uniqueness is probably what bugs people.
My UID is prime. Hah!
No, 0.999... is NOT "approaching" 1. It is a single number, defined as the limit of an infinite sequence of numbers. The numbers in that sequence approach 1. None of the numbers in that sequence are equal to 1, but the limit of the sequence is 1. The notation "0.999...", misleading though it may be, does NOT refer to the sequence, or any element of the sequence, or to the process itself. It is simply defined as the limit of the sequence, which is one number, and that number is 1.
You are being very imprecise. The LIMIT expressed by the infinite series 0.999... is equal to 1.
That's what real numbers are. One popular definition is to define them as limits of Cauchy convergent rational number series. That in turn establishes a equivalence relation on convergent rational series (namely, everything is equivalent which has the same limit). A particular subset of these rational series is the decimal representation of the number. So in other words, discussing numbers via their decimal representation is not very imprecise especially since most numbers have a unique representation.
People think that 0.999... is not 1 but that there is an infinitely small space between those two (.999... and 1).
But just keep in mind that the number 0.1 is accuraretly displayed in the decimal system, but in the binary system it is 0.00011001100110011001100110011001 and so on.
It is the same number. Just our system of displaying it cannot handle it.
1/3 (base 10) cannot be accurately displayed in base10. It can be in base 3 (0.1). Same number, still.
Thank you, Zeno.
Q: why do people who either never took, or failed, Cal I get modded Insightful?
Research is what I'm doing when I don't know what I'm doing. -- Wernher von Braun
Perhaps you should add that one to... ;)
http://en.wikipedia.org/wiki/User:ConMan/Proof_that_0.999..._does_not_equal_1
0.999... used in an equation is not actual inifinity, it is potential infinity. At best 1 potentially equals 0.999... Hardly proof!
if you want to argue infinite repeating decimals, than yes, 0.9999... is approaching 1.
That's what the expression "0.999..." literally means. There is no argument: if you consider it a finitely long number you're not even talking about the same thing.
It's limit as we approach an infinite number of decimal points would essentially make it equal to 1.
"Essentially." This is interesting. Exactly how close does 0.999... get if it never reaches it?
It gets infinitely close. The difference is infinitely small. In other words... they're equal.
Basic mistake that novices do - forget the type comparison
yes, your explanation is correct. It is the same point in the set of real numbers and just has two (2) or more different notations. One can also use different bases.
1 is not equal to zero;
0.1 is not equal to zero;
0.00001 is not equal to zero; but
0.0000...(infinite number of zeroes)...1 is exactly equal to 0
Casual inspection reveals that this must be so, as it is just 1 - 0.99999..., but you'd be surprised how many people get uncomfortable with infinitesimally small numbers being equal to zero.
~Idarubicin
If I get one of these people who can't understand it after I've tried the 'times ten' proof, I do this:
"Okay, tell me, what's 1 minus 0.999 recurring?"
"nought point nought nought nought nought nought..."
"right, keep going until you get something that's not nought. Bye"
Surely by that argiument, all numbers are rational. Pi is only a few dozen decimal places since there's no way we can measure the circumference of a circle more accurately than that.
Pure mathematics isn't a study of what's posible in the real world, but in an abstract space.
I'm not even close to a mathematician, so forgive this possibly very stupid question:
.999... by anything at all? If the sequence is infinity, then any application of task or step can never be completed as it would take infinite time to perform the calculation.
How can you multiply
I have an infinite accuracy measuring device - it's called a limit.
Doesn't exist in the real world, you complain? That's ok! The problem doesn't ask about anything that exists in the real world.
Zeno's paradox of movement is reconcilable. For your version, start by addressing the problem as a geometric series with initial step a = 1/2 and ratio r = 1/2. Then the sum s of the series is 1/2 + 1/4 + 1/8 + ..., and since the ratio is less than one, a finite sum may be obtained by the classic formula s = a/(1 - r) = (1/2)/(1 - 1/2) = (1/2)/(1/2) = 1.
If your initial step is 1/9 and your ratio is 1/10, then your series terms are 1/9 + 1/90 + 1/900 + ... = 0.9 + 0.09 + 0.009 + ... = 0.999..., and your sum is, once again, 1.
Didn't even need calculus for this one. :3
~ C.
I don't think you really get the idea of "infinity."
Heh. Classes of infinity. There are infinite natural numbers (1, 2, 3, . . . ). There are infinite integers ( . . . -2, -1, 0, 1, 2, . . . ). But they are both countable. We can say that there are the same number of natural numbers as there are integers. In fact, because you can map natural numbers onto integers (1/1, 1/2, 2/2, 1/3, 2/3, 3/3, . . . ) the sets are the same size. There are infinite irrational numbers too, but there's more of 'em :)
It may look like I'm doing nothing, but I'm actively waiting for my problems to go away.
--Scott Adams
Actually, if you define 0.999... as having an infinite number of decimal points, then it is true. And that's how that ellipsis is defined! It means exactly infinite repeating decimals.
This is a logical short-circuit, then. If there are an infinite number of 9's, then you'll never measure the difference between this and '1'. So, for all intents and purposes, assume '1'.
This doesn't seem very impressive, when you look at it that way.
How are we measuring the distance between the cat and mouse? Are we measuring from the centres of gravity? Or the skin surface? or the edge of the last hair? Are we measuring from the nucleus or the edge of the electron field of the outermost atom? If a static charge develops between their furs, does that count as having reached it? Or only if they exchange electrons?
- RG>
Hey pal, this isn't a pleasantforest, so don't waste my time with pleasantries!
Not faulty math but an artifact of the decimal notation that is used to indicate the numeric value 1=.999...
Either "1.0" or "0.999..." are notation for the same numeric quantity.
You can't sequentially write an infinite number of characters (you'd run out of time) but the notation can certainly capture it's meaning. The elements of the sequence {0.9, 0.99, 0.999, 0.9999, ...} _do_ approach 1.0 = 0.999... but never get there. So your statements are correct when applied to that _sequence_. But they are not correct when applied to the number 1=0.999...
The idea of "closeness" doesn't apply in an infinite series. The number doesn't end.
It's an abstract logical construct with no analogue in reality. Do you honestly expect an intuitive answer?
And did you just seriously say you don't care what the numbers say in a mathematics discussion? Damn son, talk about overvaluing your instincts.
Please look up the concept of an infinite series, and in particular what "convergence" means. Here is a start : http://en.wikipedia.org/wiki/Geometric_series
There is a problem with infinite decimal numbers :
0,9999... * 10 = 9,9999..., but it is harder to prove than we can think.
But still, 0,999... = 1 and there is another way to proove it:
0,99999... = sum(9*10^(-i)) = 9 * sum(10^(-i))
which is a geometric series equals to 9 * (1 / (10 - 1)) = 1
(Cf. http://en.wikipedia.org/wiki/Geometric_series#Formula)
You're describing Zeno's Paradox.
This appeared as a paradox to the classical Greeks because they had no concept of infinity or limits. Once mathematics introduced those concepts, the paradox disappeared.
Parity: What to do when the weekend comes.
In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"
The mathematician said: "Never."
The physicist said: "Eventually, they will come to a point where they would be required to move less than 1.616252(81)×1035 meters closer together. From the uncertainty principle, we know we cannot measure position more accurately than that. So either they will not move at all, or they will superimpose at that point."
The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."
Ideas want to be free.
Please share yours.
Finally had enough. Come see us over at https://soylentnews.org/
E.g., if you take basic set theory and the set of real numbers to analyse the problem:
0.99999... is the last element of ]-infinity, 1[
where as 1 is the first element of [1, +infinity[
and ]-infinity, 1[ intersected with [1, +infinity[ is the empty set...
var sig = function() { sig(); }
uhhhhh.... Ok, I like Eldavojohn, his posts are usually well thought out, reasonable, and add something to the original article. He's a good poster.
Buuuuuuuut this kinda seems like a lame abuse of wuffie. A highschool math trick isn't really slashdot material.
I guess the monty hall problem was a similar case, but that just makes this a copy-cat. And the monty hall problem was causing a stir in other places, so it kinda warranted some news. But this? meh.
At what point along your series did 0.000...9 become anything other than that? That's what your example would require.
Finally had enough. Come see us over at https://soylentnews.org/
so don't use scalar values. y^(-1) *y = 1
You're basing this on not carrying the 1, but when 0.111... is multiplied by 9, you can't carry the one because the value never ends. Then it must equal 1 based on the limit.
When you say 0.999... you can't assign a value to it, because it never ends. you can multiply 1/9 *9 and write it out in decimal form for years to come, but you'll just keep writing 0.999.... forever. Basically, 0.999.... isn't a real value, but using its limit, it is 1.
It's somewhat strange that you would use the Zeno paradox to justify that they cannot be equal; considering that the Zeno formulation is obviously wrong: in real life we have no difficult traversing distances despite this mental problem with "infinite divisibility".
But the Zeno problem is a red herring in this context in any case. The 1 = 0.99... equality is not asking about lengths of real objects in the real world. And it is not talking about performing floating-point computations on real computers that have rounding errors. It is talking about abstract mathematics. In mathematics, the definition of the ellipsis, "...", is to specify an infinite repetition of the pattern. Not a "really really long" repetition, nor some kind of temporal series where we "keep adding another digit forever" but rather it means to specify that the number has an infinite number of digits that repeat along that pattern.
There are many reasons why people can't grasp 1 = 0.99...; in your case it seems that you're thinking of 0.99... as a number where we "keep adding another digit" and then you worry that since it will take "forever" to add an infinite number of digits, there will always be some small remainder in (1-0.99...). But the mental model of infinities in terms of a temporal progression that never ends is just a crude way that humans use to think about the difficult concept of infinity. In this case, the crude heuristic leads to the incorrect conclusion. Because, again, an infinite quantity in math is not something that "grows bigger forever" it is some that simply is infinitely large. And an infinite number of digits after the decimal doesn't mean "we keep adding more digits forever" but rather than the number simply has an infinite number of digits.
Those who have trouble accepting that 1 = 0.99... should just realize that these are two equivalent ways of writing the same number. Do you similarly argue that 10/5 and 2 are not the same number? Or that 10/10 and 1/1 and 1 are not all the same number? Writing out "1" as "0.99..." or "9.99.../10" or "100*99.99.../(10^4)" may look weird, but they are all the same number, as a quick rearrangement will demonstrate.
8.99999..991
This is the mistake everyone makes. The "..." signifies an infinite sequence. There's not any digit at the end. It's not meaningful to put "..." in the middle of a decimal number and then have digits after it. (This is different from the meaning of "..." in text, which is often used to signify that some finite amount of text has been elided.)
if you want to argue infinite repeating decimals, than yes, 0.9999... is approaching 1. It's limit as we approach an infinite number of decimal points would essentially make it equal to 1. But you cannot reach infinity so this is a moot point.
I have no idea where to start with parent's post, but I'll just deal with this.
The string of characters ".9999..." *means* the limit of (9/10 + 9/100 + ... 9/10^n) as n tends to infinity. The partial (finite) sums form a sequence that indeed converges to a number, which is defined to be the limit of the series, and that number is 1... the value of the given sum. What does "converge" to 1 mean? It means you can get arbitrarily close to 1 by going out sufficiently far in the series.
Also, I'd be curious to know what you think the result of (1 - .999...) would be.
Look, there's a very simple question that solves this problem:
What number would you add 0.99999... to make it equal to 1?
You would add 0.000....001 to it, of course.
Except that's an infinite number of zeros.
Which means that this number is infinitesimally small - literally, it would have to be the smallest number greater than zero (in other words, it's epsilon).
So we can re-write the equation as 0.999... + epsilon = 1.
However, epsilon doesn't exist. There is no smallest number greater than zero.
Which means that in order to make 0.999... + epsilon = 1, you have to use a non-existant number.
Which means that 0.999... = 1 already.
Nope. 0.999... repeats infinity. the 9's never end. multiplying by 10 moves the decimal over 1, but doesn't add a 0 at the end of the value.
Hey Rocky, watch me as I pull another significant digit from my hat!
Showing 0.999.... at the beginning implies that the number is NOT 1. And then at the end, it's turned into 1.
0.999... has infinite significant digits. It's how the trick works. Otherwise you have to say that:
10a != 9.999...
10a = 9.9
This isn't insightful, it's wrong. Painfully, painfully wrong.
Regarding parent: I see the Zeno thing; but this analogy is not the best... the cat only has to get within about 10cm. :)
Which reminds me of a qn that's bugged me for a few years: Are infinites only theoretical constructs?
If infinites can exist in theory but not reality, then 1 == 0.999... (an infinitely long number) only in theory, but never in reality.
Is this just a neat way to introduce elipses to 3rd-graders, or do any other results in mathematics depend on this?
then the cat stretches his neck a bit and eat's the little smart-ass
Snowden and Manning are heroes.
huh? the 9's never end. So anyone wanting to add a value at the end of 0.999.... (represented with ANY amount of places after the decimal point) is just wrong. There is no end. So basically you argued that 0.999 DNE.
Interesting. Thanks for that. Maths was never my thing.
Thank god the cat doesn't know that either.
Finally had enough. Come see us over at https://soylentnews.org/
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders a half a beer. The third orders a quarter of a beer. The bartender says, "You're all idiots," and pours two beers.
In order for any two rational numbers to be different from each other -- for that difference to have any meaning -- you need to be able to define a third rational number that is in between those two.
You can always find a 3rd number in between any two rational numbers. In fact you can find an infinite amount of values in between any two rational numbers. All you have to do is keep adding another decimal.
Except 0.999... is not a rational number. It is an irrational number. There is no 3rd value you can define that is between 1 and 0.999...
Which means any difference between those two values is meaningless.
I prefer the integral. That way, you can build one out of electrical components.
You're right, it isn't approaching 1, it is 1. But that is the definition of the limit. lim_x->9 of x/9 = 1
Your mind scream "bullshit!" because it is "bullshit!". This 'proof' is just a proof that the decimal system doesn't work for all numbers. It just shows a rounding error that is infinitely far down the line. It is still a rounding error introduced by the decimal system.
It's ok, I've been schooled enough. Calculus, infinite series, convergence, Limits, Zeno's Paradox. Thanks all!
Finally had enough. Come see us over at https://soylentnews.org/
DAMN IT Wrong Thread.
Letter To Iran
What happens when x == -1?
-----------
100% pure freak
When studying EE way back when, we generally worked with +/-10% in our circuit equations but a chemical engineering student friend of mine pointed out that chemistry was somewhat less precise than electronics and they used +/-30%. Thus 2.3 + 2.3 = 4.6 which rounds to 5 so 2+2=5 for sufficiently large values of 2. ;-)
I've read the first few pages of the PDF, and the paper, while presenting a few interesting tidbits about mathematical research (e.g., a semiring where .(9).(9)=1, "so long as the number system has not been specified explicitly, the students' hunch . . . can be justified in a rigorous fashion." It is true that in a number system other than the field of real numbers (which, of course, includes the completeness axiom), .(9) may be not equal to 1. However, there are a number of other things students deal with at the same time that require that \mathbb{R} be the field/metric space/algebraic structure under study. One of the obvious ones is that 1/3+2/3=1 when, in fact, without the completeness axiom (and other things that necessarily make .(9)=1 in \mathbb{R}), we would have .(3)+.(6) and have no way of actually showing this equals 1.
So the paper is interesting for its idea that the reason students don't understand .(9)=1 is because they're not taught about Cauchy sequences, fields, limits, and the axiomatic structure underlying \mathbb{R}. However, it does make a few weird statements in its discussion.
And, in my opinion, students don't understand .(9)=1 simply because they refuse to understand the simple fact that "if something is proven true, then it is true no matter what you think to the contrary [unless you reject the axioms, and you should be prepared for the consequences if you do]." When I learned the x=.(9); 10x=9.(9); 9x=9; x=1 proof when I was in elementary school, my reaction was "holy crap that's awesome."
The problem, I think, is willingness to trust mathematical proofs over base intuition.
Actually, if you define 0.999... as having an infinite number of decimal points
Mathematics grammar nazi says: :-)
"It has an infinite number of decimal places, but only one decimal point"
Not confused enough? http://translate.google.com/translate?u=www.slashdot.jp&hl=en&ie=UTF8&sl=ja&tl=en
[snip] irrational [snip]
You keep using that word. I do not think it means what you think it means.
No, you're missing the whole point. 1/3 is exactly equal to 0.333... with an infinite number of trailing digits. It's not an approximation or an estimate, it is two ways of representing the exact same real number.
Here's how you convince yourself: If 1/3 was really close but not quite 0.333..., then we could split the difference between those numbers and find another real number between them. But we can't, which means we were wrong to assume that 1/3 and 0.333... were distinct.
I remember learning about endlessly repeating numbers back in like 2nd grade! Repetend FTW!
So have you heard about these prime number things? Pretty cool!
If the only way you can accept an assertion is by faith, then you are conceding that it can't be taken on its own merits
It's more fun to work out why this proof fails when using non-standard analysis (in which 0.999... != 1).
Quidnam Latine loqui modo coepi?
^h^h^h^h^h^hcarrier lost
modems in Montana.. now that I believe
01010010 01100101 01100001 01101100 00100000 01101101 01100101 01101110 00100000 01110101 01110011 01100101 00100000 01100010 01101001 01101110 01100001 01110010 01111001 00101110
--
Congratulations Fry, you've snagged the perfect girlfriend. Amy's rich, she's probably got other characteristics...
SJW: Someone who has run out of real oppression, and has to fake it.
I was taught this 15 years ago in High School math class (or was it 20 years ago in Algebra...), and it wasn't even close to new back then. Are we going to see slashdot stories talking about a wondrous proof about the relationship between the sides of a right triangle and its hypotenuse?
i.e. Think of pi.
you have NEVER done math with pi. ever. without question.
you have only ever done math with an approximation of pi. there's a BIG(small) difference.
To which the mouse sits down gently, safe in the knowledge that he will never be caught by the cat. After all, no matter how close the cat gets, he can only get half the distance closer with each step....
And after taking about 6 steps, the cat came within chomping range of the mouse.
Chomping Range = Distance Traveled + Distance from Foot to Teeth
Exactly.
He should have stopped at 95%. Everyone knows the last 5% of a drink is saliva, so everyone past the 10th or so isn't getting any beer at all.
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Sorry, but as a graduate student in math, I can't agree with this.
.11205344344344344344... Then 10^8 x = 11205344.344344... and 10^5 x = 11205.344344344... Hence 10^8 x - 10^5 x = 11194139 so x = 11194139/99900000, a rational number.
Firstly, your terminology is wrong, infinity is not a number, much less an irrational number. An irrational number is defined to be a real number (loosely anything that can be expressed as an infinite decimal) which is not a rational number (a fraction). So infinite repeating decimals are not irrational. Their infinite repeating nature allows us to perform a trick similar to one mentioned above where we multiply by a suitable power of 10, subtract the original times a lower power of 10, and divide to get a rational number. Take for example x =
Only numbers with infinite non-repeating decimal representations (like pi or e) can be irrational.
The spirit of what you are saying makes since - when we start to deal with infinite numbers strange things start to happen. However it appears that you lack the necessary background in analysis to understand what those strange things might be. The idea behind this is that an infinite decimal is actually a sum of infinitely (specifically countably) many rational numbers which converges. That is, they can only be viewed and examined with the technology of limits. Within the topology normally associated with the real numbers, the multiplication function is continuous, and hence interchangeable with limits. In particular, this means that the multiplication operators act on infinite repeating decimals in expected ways. So in this particular example, these "different things" that you claim happen really don't happen.
Base 10 is flawed. Our math-system isn't accurate enough because we all know 0.9999999999 1
An infinite number of mathematicians walk into a bar. The first orders one beer, the second orders 2, the third orders 4. The bartender says "If you keep that up, you'll end up owing ME a beer"
Do you raise it up? Or wax is down?
I am from the US, and I learned the same thing at about the same age.
As poor as the US school system in general has become in the ensuing years, there are, I am quite sure, a large number of US 12 year olds learning this as I write.
Mod parent up
This is actually much better proof than any formulas. Just go back to math definitions and it's there.
It took me some time to recall math class and formulas we used to prove it.
dx=an infinitesimally small number that is greater than 0, as would be defined for a function f(x). If you subtract dx from 1 you should have a number that is less than 1 but as close as possible. dx approaches 0, but dx never gets there. If dx was 0, you would not be able to calculate the slope of a line, dy/dx. In fact 0.999... should be equal to 1-dx
The real question is, doees it work under a different base. IE base 2, 8, or 16.
If someone can prove 0.FFFFF....x = 1x I will be happy.
ASSUMING DIRECT CONTROL
There is no -1 disagree
Look at the following: 2x > x So what happens when "x" is set to infinity? We know that "2x" is always greater than "x", but since infinity is an irrational number, different things happen.
No. Nononono. It hurts.
Infinity is not a number! You cannot 'set x equal to infinity'. x as used is here a variable which can contain some real (or perhaps complex, depending on how you look at ">") value, but it cannot contain a concept! That would be akin to setting x equal to addition or something like that.
Yes, in mathematics, we sometimes talk about an "infinite number of n" or a "set of infinite size", but this is a simplification which is applying the concept, rather than trying to count up the number of objects and saying that it is "equal to infinity". [Yes, I know the difference between countable and uncountable infinities. It is irrelevant -- "countable" is again a shorthand for a concept.]
We know that "2x" is always greater than "x"
And if x is less than zero?
>> Standing on head makes smile of frown, but rest of face also upside down.
Some women like java but I find the ones who like vodka are much easier to impress
GP poster was making a clever reference to the the song Montana by Frank Zappa. The song is about a man who moves to Montana to grow a crop of dental floss.
If you are over 14, not knowing this has reduced your Geek Cred rating one level.
Some mornings it's hardly worth chewing through the restraints to get out of bed.
Last time I checked, ( a few seconds ago) .999 times 10 = 9.99 NOT 9.999
Maybe that mathematician needs a refresher course. .999 times 9 = 8.991 + .999 = 9.990 or 9.99 for short
- A Frog in a pond utters an azure cry. -
The problem with the argument you present is that people who don't believe 0.999...=1 also don't believe that 0.333...=1/3. They can't quite wrap their heads around the concept of infinity, so in their minds 0.333... continually comes closer to 1/3, but never quite reaches it because they can only imagine a finite number of digits. They honestly think of infinity as being a really large finite number, so they believe that no matter how many digits you add to 0.333..., it never quite reaches 1/3.
Another part of the problem is that many people simply can't wrap their heads around is that they don't separate the idea of a number and the symbols used to represent numbers, thus they cannot grasp that some numbers can be represented in more than one way by our number system.
Your cat and mouse example is a perfect example of exactly why 0.9999... = 1! Cat and mouse are 1 m apart, cover half the distance, 1/2m, 1/4m, 1/8m...infinity later, and sum them all up, and this is equal to exactly 1m. The reason the cat cannot reach it is because it DOES require infinite steps, and the cat cannot take infinite steps. However, 0.9999999.... has infinite digits already. You cannot have a handful of 0.9999.... cups of something, because it requires infinite steps to produce. But on paper, you can have such a number.
Put another way, if 1 is greater than 0.9bar, then what is 1 - 0.9bar? 1 - 0.9 = 0.1, or, ten to the minus one. 1-0.99 = 0.01, ten to the minus two. AKA, one minus (zero followed by x nines) = ten to the minus x. How many nines, again? Infinity. OK, there is your answer. The difference here is ten to the minus infinity. How do we handle that power? Who cares. What does it mean? It means infinite zeros followed by a one. In your example, the cat cannot reach the mouse because, if the space is divided into an infinite number of "allowed" steps, it cannot cover all infinite steps in its finite lifetime. By the same argument, if there are an infinite number of zeros, there is no one at the end. You with your infinitely accurate measuring device would spend your entire life SURE that if you just keep going, it will eventually say "1", but it won't, not ever. 0.0000000000000000000000000000.... is just 0. 1 - 0.9bar = 0, so they are equal.
ASCII stupid question, get a stupid ANSI
Legitimate mathematical philosophies of finitism or ultrafinitism invalidate any 0.999...=1 proofs before they can even begin.
Welcome to the cutting edge of mathematics 2500 years ago.
Infinity gets weird as hell. I don't remember the exact proof, but in a calculus class in college, the professor posed this question for "fun": if you have a ball that recovers 9/10 (or whatever the figure was, it's been a while) of its height with every bounce, how long will it take it to come to a complete stop? One of the students worked it out before the next class session, and we were all amazed to learn that even though the ball will make an infinite number of bounces, it will do so within 30 seconds. So an infinite number of actions can, theoretically, be performed in a finite amount of time. Pretty crazy, but the math was sound.
"16MB (fuck off, MiB fascists)" - The Mighty Buzzard
Now that I see they meant .999... to infinity the calculation would be .999... * 10 = 9.999...0 emphasis on the zero at the end. While you can never get to the end of an infinite number, if you could you would be required to add a zero after multiplying by 10
- A Frog in a pond utters an azure cry. -
But the idiots who modded you informative should really have known better. Factually wrong statements are not informative and if you can't tell right from wrong, you shouldn't mod.
No one said anything about them not being able to tell right from wrong. They may have simply been misinformed. People are firmly convinced of the truth of things which are wrong all the time, you can't expect someone to go out and double-check everything just in case they happen to be wrong on this one.
"16MB (fuck off, MiB fascists)" - The Mighty Buzzard
You got your Computer Science in my Math!
... + (1/9) ten times. Or are you saying that we can't add either?
.999... This notation implies the existence of a limit, and so attacking this problem with a tool that has only finite precision will necessarily be flawed. It must be approached with the tools of analysis, and once these tools are understood, the proof is trivial and the fact is obvious.
But seriously, since when do we define multiplication by how accurately we can perform it with computers? There is absolutely nothing wrong with multiplying (1/9) by 10. If I recall, the answer is 10/9.
Indeed, multiplying by positive integers at the very least has an easily definable (even in CS) meaning. Just add that many times. So 10 multiplied by (1/9) would be the same as (1/9) + (1/9) +
As many others have said, this mostly comes down to the problem of definitions. People have trouble accepting that this is true because they do not understand the notation
Please note that the first part of his theorem stated a = b. Please try to read ALL of the message thread before commenting. Thanks.
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You're not measuring anything, of course. You're defining something and forming a conclusion from rigorous logic. There's no "intents and purposes", either, it actually is exactly 1 as defined.
No kidding!!! What do you say at this point?
Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1.
This is exactly the kind of irresponsible use of the 'universal language' that if it were put on the side of a space probe would doom humanity to annihilation.
There's no place like
Depends on the people.
I really like the Monty Hall problem, but I figured out how to explain it in a way which makes it easy for the light go on. That's fun for everybody, (well, those who enjoy such problems. Other people just glaze over). But yeah, I confess, I was argumentative the first time I encountered it until I worked it out a day or two later.
-FL
As soon as you get to "You know you can represent 1/3 as 0.333... right?", you hit a brick wall. People who believe that 0.999... does not equal one also believe that 0.333... does not equal 1/3, and for many of the same reasons. Taking your approach, you simply shift from arguing about whether or not 0.999... equals one to arguing about whether or not 0.333... equals 1/3. You have to get at the root of the problem of why they refuse to believe those numbers are equal before you can get anywhere.
The problem i see is that in presentation of the problem we are shown an endless series: ie: 0.999... Notice the elipsis ..
Then, suddenly, the 'endless series' disappears, and is presented as: 0.999
Those quantities are not identical.
And therein lies the fault.
"There are 11 kinds of people: those who know binary, those who don't, and those who could not care less!"
This particular use of rigorous logic serves no useful purpose.
but in maths we say that:
limit of n/(n+1) with n to infinity equals 1.
And as usually with maths, common sense also works :-)
I've been in plenty of arguments about this on game-related forums, and the answer I get to what you propose is zero, followed by an infinite number of zeroes, which itself is followed by a one. People who don't understand that 0.999... = 1 also don't understand that you can't have any digits after an infinite number of digits because they don't understand what infinity means. They honestly think infinity is simply a very large finite number, therefore they think it is possible to have an absurd number such as 0.000...1.
Are your 'y' and 'o' keys broken or something?
sic transit gloria mundi
The error comes at step 2 (or even step 1). We just mentally brush under the carpet all the other 111s stretching off to infinity. But what does it mean to multiply an infinite set by 9? Imagine we get a computer to do it for us. So we have a program to multiply our infinite series of ones by 9. It will never finish of course. And there's a big difference between finishing and never finishing. Multiplying by 1 is an atomic operation. Multiplying by 0.999... is a process that will never complete. In fact the number 0.999... is itself a process, not a fixed quantity. I wonder if maths could be recast in terms of processes? So instead of saying for instance the sum of 1 + 1/2 + 1/4 + ... == 2, you'd leave it as a process which would be a bit more awkward but maybe those process numbers would combine/cancel out.
.999... is not a rational number, it's a real number.
Wrong, as the GP said:
... all numbers that have "infinite repeating decimals" are rational.
This is just like 1.0 which is a rational number. In computers the integer 1 is different than the floating point 1.0, but rational vs. real is a mathematical concept and in math 1 = 1.0.
People who do not believe 0.999...=1 also will not accept that 1/9=0.111...
Since you are beginning your proof with something they do not accept as true, they will not accept your proof.
Therefore 0.999
and 0.999
so with an infinite number of comparisons, 0.000...1 == 0.999...
and since you can multiply any number in the range 0 - 1 by something to get every other number, logic would indictate that all numbers have the same value PROVIDED the initial assertion was true,
politicians are like babies' nappies: they should both be changed regularly and for the same reasons
Does 0.9998... also =1?
Mathematicians berate and scold engineers who use the concept of time=0+ or 0- to denote you are referring to just before or just after an event, now it appears they've come up with a proof that we're wrong...
Jokes on them, I still make more money using my disproven notion than most of them ever will. Hah!
For the same reason that ...
1 = 1/2 + 1/4 + 1/8 +
As they converge to infinity. they both sum up to 1.
The difference between them is infinitely small.
I don't know the meaning of the word 'don't' - J
Either you misunderstood your physics teacher, or your physics teacher doesn't know math. (Not an uncommon ailment among physicists)
Le français vous intéresse?
The really interesting part of the paper was about "non-standard" analysis where there are infinite numerals between terminating numerals, eg "0.999...999" I'd never heard of that construction before. They mention how Leibnitz wrote of trying to imagine of a line whose length is longer than any finite line, but which also has end points. Mind bending.
I get that, however, once you multiply it by 10, the resulting number has (infinity-1) decimal places, not infinity. I know the theories of math say that infinity - 1 = infinity but I don't buy it as infinity will always be greater than infinity - 1.
yes. the correct representation is .3 with the little line over it. .33, .333, .3333, .33333 are all incorrect. These are not 1/3
it is laziness. Same with the guy up above who just tried to prove that -1 = 1. He threw away the signs in one step for no reason and created a false proof. pure laziness attempting to pass as clever intelligence.
Mathematical proofs are a way of finding new properties of a system by making deductions from previously known properties, and in a practical sense are often a short-cut finding, the new property without testing every possible case.
For a simple example, consider the property that every integer multiplied by by 10 will end up with a zero in the ones place. Someone could respond: "How could you know that? There are an infinite number of integers and it would take infinite amount of time to multiply each by 10 to check it." But using a proof can rigorously show this is a pattern without testing every number by exploiting the properties of numbers.
In the case of multiplying 0.999... you can workout what the pattern any given digit will follow, and use that instead of manually performing the calculation.
Did you just try to put digits after the infinite number of nines? You are correct that there is something idiotic going on here, but I do not think you understand what that something is.
Sorry, I think you're one of the people Benfea is talking about.
0.0000...1 is not a real number. You can't have anything after an infinity because then it's not an infinity. Infinity goes on forever. If you put something after forever, you don't really have forever in the first place.
a = b
ab = b
ab - b = a - b
b (a - b) = (a+b)(a-b)
b = a + b
b = b + b
b = 2*b
1 = 2
From an AC reply:
No, since 0.0000....1 represents a number that approaches zero, but isn't zero. If you use that number in a division, you get infinity, as opposed to an undefined or indeterminate result.
you have only ever done math with an approximation of pi.
This is only true if you define limit "math" to mean arithmetic and what simple calculators do. Algebra gives the abstract tools to work with numbers without needing the decimal expansion. By trigonometry and especially calculus, pi gets used a lot in an exact sense. Although sometimes the fundamental basis of what it means to work with an real numbers doesn't get covered until a course on real analysis.
So the difference between 1 and 0.999... is an infinite number of zeros followed by a 1
1.0 - 0.999... = 0.000...1
If the amount of actions are infinite, then they never stop. It would be logical fallacy to say an infinite amount of bounces will *ever* complete. If it stops bouncing, than it can logically be concluded that an infinite number of bounces had not occurred.
Slashdot will not accept the bar over the top. A trailing "..." as used in the summary is also acceptable.
See my journal for slashdot ID's by year. Mine created in 2005. http://slashdot.org/journal/289875/slashdot-ids-by-year
Numbers in general are mental constructs so your question is difficult to answer (at as you have phrased it). For example, the number "3" isn't a tangible object. Many things are modeled by the number 3 (e.g. the number of strikes to strikeout a batter, the number of cookies in a jar), but those things are only modeled by "3". They are not "3" itself.
If we rephrase your question as "Is there anything that infinity is a good model for?", then the answer is yes. For example, if you accept the idea that a perfect circle can model real things, then the ratio of the diameter to the circumference also models a real thing. Of course, this ratio is pi, which has a infinite number of non-repeating digits. Accurately modeling ballistic paths or planetary orbits essentially requires summing over infinite number of infinitely small steps (i.e. integral calculus). Taylor series expamsions have an infinite number of terms but form the foundation of modeling waveforms including video and audio compression. In computer science, modeling a recursive function or proving properties of that function often requires an infinite expansion of that function (a.k.a. co-induction). Interactive input to a program is also best modeled by assuming that sequential input is stored in an infinitely long list.
Some of these examples, may not seem like "real" infinites to you, but that is because the common notion of infinity as a "number bigger than any other number" is misleading. Infinity is a modeling tool for certain problems where we would normally like to count something (e.g. the digits in pi), but where we aren't actually allowed to stop counting (e.g. there is no last digit of pi).
(Qualification: There is a branch of logic (i.e. Finitism) that reject infinity as a usable mental model, but until you understand the difference between constructive logic and classical logic and their relation to the rule of excluded middle, you should really ignore finitism. These are doctoral level math/logic topics and for anything at a lower level than that it is actually simpler to use infinity as a mental model than to try to live without it. At that level you can choose between a number of different mental models.)
I'm not missing the point. This proof just exposes a flaw in how we currently envision and deal with infinity. Just like infinity * 0 being undefined when it should logically be zero. Understanding multiplication to be a series of a additions this becomes painfully obvious. No matter how many times you add nothing to itself you will still have accumulated nothing. No matter how big a number you think you might start with, if you never begin writing the equation, you still have nothing.
I swear some theoretical mathematicians really just over-complicate things for no good reason.
You're wrong. It is logically true, and provable, although I cannot render the proof offhand (maybe someone else can).
Let's put it into another context which may help. The space between 0 and 1 is finite. We can clearly delineate the start and end. Yet, in that finite space, there is an infinite number of real numbers. This is something like the "infinite bounces in finite time" trick. It's true that in the physical world there aren't an infinite number of bounces, because outside factors will interfere and stop the ball from bouncing prematurely. However, it is possible in pure logical terms.
"16MB (fuck off, MiB fascists)" - The Mighty Buzzard
I'll try this approach the next time I have to explain that problem to somebody.
Tell it to my bankers..
"Computers are a lot like Air Conditioners" "They both work great until you start opening Windows"
I have eleven fingers. I hold up both hands and count down- ten, nine, eight, seven, six. Plus five on the other hand equals eleven. So 10 = 11.
This works in any simple base using the same concept of decimal representation.
In base x, consider the number zero followed by n digits of (x-1) after the decimal point, e.g. 0.FF...F with n Fs for hexadecimal. One minus this number gives the difference 1/x^n. In the limit n goes to infinity, this difference goes to zero for real numbers. And with the real numbers, zero difference means they are the same number.
Women = Time * Money.
Time = Money.
Women = Money^2.
Money = sqrt(Evil).
Women = Evil.
So you are saying there is a school of thought that says there is no such thing as one third?
ASCII stupid question, get a stupid ANSI
Infinity is a potential number it is not a number. I have a very good concept of infinity. I just choose to disagree on some of the points used when dealing with it.
I'm no mathematician, and have long grappled with the problem as stated. But I recently came to a conclusion: mathematics means what we want it to mean, regardless of where "proofs" take us. Consider the following: .999... = 1.000
but
6.626 x 10**-34 (planck's constant) != 0
How can this be? The first number approaches 1
but never reaches it, and we declare that it is
in fact 1. But the second number is so vanishingly small and so near 0, and we declare that it is NOT 0 (because it has been useful so far). There's a large inconsistency here.
I think mathematicians and physicists are trying to have it both ways because they do not want to confront the contradictions in their math and thus in their worldview.
Imagine that. The standard model of physics could fail if traditional math fails. People might lose their grant money because their science is built on a house of cards (mathematics). Horrors.
Mathematicians are distinctly countable.
You're missing the point entirely. How about I make the statement that 1.000(0) != 1 since there is an infinitely small rounding error.
If the rounding error is as you say "infinitely far down the line", it doesn't exist.
I'd tell a UDP joke, but you may not get it. I'd tell a TCP joke, but I'd have to keep repeating it until you got it.
You know I like the shirt that say 2*2=5 (For incredibly large values of 2) as well. It does open minds for possibilities outside of the normal but it is not solid math nor should it be considered to be. A number with an infinite amount of decimal places cannot be defined and thus does not exist. Trying to prove that an integer (which does exist) is equal to an imaginary number which does not exist is both pointless and asinine.
That's easy. You just use
static { ...
}
The proof is correct as are the others (1/9 = 0.111111..., multiply both sides by 9, simplify fraction on the left, 1 = 0.9999999
Sorry 1/9 is approximately = .111.... it is not equal to it. Just like pi is approximately equal to 3.1417 (to however many decimal places you need at the moment). These numbers, however, are not actually equally to the decimal representation of them which is why people use the symbols pi, e, i, etc. in their place.
But an infinite number of zeros isn't followed by anything, as there is an infinite number of them.
No, but let's not even go there.
You're lucky ; I was also twelve when I heard of this, and I was so fascinated I wanted to show this proof to my math teacher. He just flatly told me that I was wrong!!
a = 0.999
10a = 9.990
10a -a = 9.990 - 0.999
9a = 8.991
a = 0.999
"Never attribute to malice that which is adequately explained by stupidity." - Hanlon's Razor
so according to you, in the set of integers, 1=2 because there exists no x where 1 LT x LT 2. That is just stupid.
You're ignoring the possibility that some *irrational* number exists between 0.9999... and 1. In general any number of irrational numbers exist between any two rational numbers, even if there isn't enough space for "a single mosquito fart".
The proof cited it he summary isn't really a proof, it's more of a demonstration that we really don't want 0.9999... to be any different from 1 -- not if we want the normal rules of algebra to make sense. Since we're talking about a question of *notation*, it's enough to show that people who want "0.999..." to mean "1" don't have any fancy explaining to do in this case.
If you want "0.999..." to mean something else ... well you *can*, but you've either got to (a) exclude "0.999..." from the normal operations of algebra or (b) come up with some kind of extension to algebra (like imaginary numbers) that is self-consistent in all cases such as that illustrated. It wouldn't be the end of mathematics if somebody did that, but of course nobody has.
I think the real problem is that people who want "0.999..." to mean something different than "1" haven't figured out what that something is.
Post may contain irony: discontinue use if experiencing mood swings, nausea or elevated blood pressure.
A number with an infinite amount of decimal places cannot be defined.
On the contrary, it can easily be defined.
thus does not exist.
No numbers actually `exist' - they're all an abstraction, even the integers.
anytime im presented with the Monty Hall problem, I only switch 99.99999...% of the time
You must be using some other form of logic that I am not aware of. You are talking about a bouncing ball, that is a physical object. If it is going to bounce an infinite number of times, it will take an infinite amount of time because it will never stop, ever. Hence, infinite amount of bounces. Once you cede that the ball will stop bouncing, you have also ceded that it did not bounce an infinite amount of times.
The math may have involved infinity in the terms of as the number of bounces approaches infinity, but not an infinite amount of bounces.
obviously 0.000...01. Or as undefined in math. It certainly wouldn't be zero.
You're confusing mathematics with computations on a calculator. On a pocket calculator, you really cannot compute with pi, because it only computes with limited precision floating point numbers. On the other hand, using a computer algebra system which does symbolic computation, you can even do computations with pi.
Also, you can quite easily do mathematics with pi. How else do you think people prove things like sin(pi) = 0?
Mathematics is not the same as doing computations. In fact, the two things have been different at least since Euclid published his Elements. The relationship between the two things is that some parts of Mathematics provide the foundation for doing Computations, and many (most?) parts of mathematics make heavy use of symbolic computations.
I bet that SOB bartender short pours the second beer.
no, they are so close that we can't comprehend it. But there is a space. In dealing with our physical world on this planet, this rounding off (so to speak) of infinite smallness works fine, gets us close enough to be considered true. But applied to gargantuan numbers like the size of the universe, it may become pretty significant.
on my vintage Intel Pentium P5...?? http://en.wikipedia.org/wiki/Pentium_FDIV_bug (for those too young to get the reference)...
Sometimes the light at the end of the tunnel is the headlight of an oncoming train.
People who believe that 0.999... does not equal one also believe that 0.333... does not equal 1/3, and for many of the same reasons.
For once in my life I can claim someone is underestimating the average person!
I don't believe .999... = 1. Let me qualify that a bit, I intellectually and academically know it, but on a softer, more psychological level, I don't actually believe it. When presented with it, my first reaction would be "Hell no! Stupid.", even though I know it is true.
Why? Because your mapping two concepts that we all were taught as a kid isn't true. Does .9 = 1? Or .99? Or .999? or ... Or .999999999999(a ridiculous but non-infinite number of times)? Most grade school kids would say "no", and be correct. Then you hit the infinite jump, and suddenly it becomes true. So you run into two problems, the problem of it not being immediately obvious (common sense), and the problem of conceptualizing infinity.
On a lower level, its like saying A = ~A. You have a proof saying basically that ~A was A all along, so the actual preposition was wrong, which makes sense, but on a surface level all you can see is A =~A.
I have no problem whatsoever with 1/3 = 0.3333... This makes sense, its like stating A = A. 1/3 being 0.3333 is obvious. I would even get in trouble in lower level math classes for not mucking with fractions, and going straight for the decimals, since I never say fractions outside of cookbooks and socket sizes. 1/3 = 0.33333... makes sense, it is clear and obvious, and can be explained with a single phrase (not a proof); "the "/" means division". .999999... doesn't have this.
No, I'm not stupid, or at least for this reason. I know damn well that 0.9999... = 1, and if I ever find myself in a situation where that bit of knowledge can be applied (usefully, not just for building my ego on the internet), I will do it properly. My first reaction is still "bullshit!" on a visceral level, though. I don't perceive it as true, even if I know it is.
I suppose I can map this experience to most of the "social knowledge vs. science" debates in our culture currently. I won't.
A patriot must always be ready to defend his country against his government. -edward abbey
I get that, however, once you multiply it by 10, the resulting number has (infinity-1) decimal places, not infinity.
No, it doesn't. There's an infinite number of 9s, and thus moving any finite number in front of the decimal place still leaves an infinite number after the decimal place. You never run out of 9s after the decimal place in either case.
I know the theories of math say that infinity - 1 = infinity but I don't buy it as infinity will always be greater than infinity - 1.
It may seems intuitive that "infinity - 1 < infinity", but that's based on the understanding that infinity has some "value" and that you can subtract one from it. But infinity is not a number, and "infinity - 1" is not meaningful in the theory of math. Think about it: What number can you add 1 to and get infinity? There is no such number. So you have to think of infinity in a little different way, because it is not a value, but rather the concept of "without end". And if something is endless, taking some finite amount out of it still leaves it endless.
Try this: If you have an infinite conveyor belt that provides a never-ending sequence of beer bottles, does plucking one beer off the belt cause the sequence to end? No, it's still never-ending, which is what infinity means. After any finite amount of time, it's true that the number of beers that have gone past you on the belt is one less than it would have been otherwise, but after an infinite amount of time, the number of beers that have passed you is still infinite.
But you can never reach that result by just counting the number of beers at any given moment, and waiting until the numbers are equal. Because that would take infinite amount of time, and then you're no longer dealing with a number of beers, but rather infinite beers. That's the difficulty of thinking about infinity -- you can't think about it in terms of doing a finite amount of steps and waiting for it to tick over and become "infinity".
By the way, there are in fact different "sizes" of infinity, countable and uncountable but I don't want to go into that. :)
The enemies of Democracy are
How can you multiply pi by anything at all? After all, not only does it have an infinite number of digits, we don't even know what they all are! .333...
How can you multiply 1/3 by anything at all? 1/3 =
How can you multiply 1 by anything at all? After all, 1 = 1.000000... which is an infinite number of 0s.
There are other prrofs for the .99999(...)==1 argument, how about trying the geometic series one? The sum of the infinite series ".9+.09+.009+.0009+..." can be calculated from the formula "SUM==(initial term)/(1-common ratio)", or "SUM==(9/10)/(1-(1/10))", or "SUM=="(9/10)/(9/10)". It gives the exact value (1), not an approximation.
-Space for rent
It's not just about alternative representations, I think you've missed the point. Also, it makes much more sense if you're in a classroom with calculators. The kids see the calculator representations, and that 1 / 3 = .33333333 and then * 3 = .99999999
What you're addressing is partly the perception that a calculator can represent answers correctly. A perfect calculator would include the 'repeating' designator, so .3 with a line over it instead of however many decimal places you happen to display. If you always do pencil math, and never represent numbers in decimal which cannot be expressed in decimal, your rant is sound. But irrelevant since you won't have to deal with this.
Nevertheless, it's a standard operating procedure to present a puzzle and then use reason and logic to work your way out of it. In fact the entire point is to present something that probably doesn't make sense to most people at first. It is a great introduction to the concept of infinity. .9 is not the same as 1, .99 is not, .999 is not. No matter how many times you add a 9 to the end, it's not equal to 1. But adding an infinite number of 9's to the end makes it exactly equal to 1. That is what this is trying to teach, although most teachers don't go into that at the same time unless the students ask.
This is a special case of the "representing numbers in different ways" concept, and hopefully afterwards students can mentally translate between seeing .333333 on a calculator to the representation 1/3 instead.
I'm surprised you've taken a course on discrete mathematics and have never heard of a geometric series.
http://en.wikipedia.org/wiki/Geometric_series#Proof_of_convergence">http://en.wikipedia.org/wiki/Geometric_series
Take a look at that = sign. That's right, the infinite sum is EQUAL to (not approximate to) 1/(1-r). For your example, the sum is equal to 1
Try it out, 1/2 + 1/4 + 1/8.... you're going to end up with .999...
But it can be proven the sum is equal to 1. Guess what? .999... = 1.
/3. You have to get at the root of the problem of why they refuse to believe those numbers are equal before you can get anywhere.
And that is the difficulty in contemplating infinity. Running decimal places out to infinity doesn't always compute...the average person rather thinks of a very large number of places. Some think of this large number getting larger. But it's not a natural mental concept to contemplate infinity.
Get your facts first, and then you can distort them as much as you please.--Mark Twain
e^{i *pi} + 1 = 0
That is an example of performing math with pi. Just because pi does not have a sensible representation in the base 10 number system does not mean it does not exist as a precise number.
Furthermore, 10 (base pi) is exactly equal to pie in just 2 small digits.
You are talking about a bouncing ball, that is a physical object. If it is going to bounce an infinite number of times, it will take an infinite amount of time because it will never stop, ever. Hence, infinite amount of bounces. Once you cede that the ball will stop bouncing, you have also ceded that it did not bounce an infinite amount of times.
No. It will take a finite amount of time. That does not change the fact that infinite actions can take place in that amount of time. It's important to remember that it is not merely the height of the bounce that shrinks, but also the time it takes to bounce that shrinks. Because the time for each action is approaching zero, infinite actions can happen in the finite amount of time.
The math may have involved infinity in the terms of as the number of bounces approaches infinity, but not an infinite amount of bounces.
Those are the same thing. If the number of bounces approaches infinity as t approaches 30 seconds, it means an infinite number of bounces happen in those 30 seconds. You're drawing a distinction that doesn't exist.
"16MB (fuck off, MiB fascists)" - The Mighty Buzzard
but symbolic computations are just that: symbolic.
you cannot represent much of out mathematics in the real world, which is where computation plays so wonderfully. give me a real world example of sin(pi)=0
I can tell you for sure, that no matter how you experiment, and no matter how accurate you think the answer is, it's not true in the mathematics sense.
the value of pi that we determine is based on the idea that a perfect circle exists. as we know from nature, no arc shaped structures really exist (at the atomic level, they much more closely represent a Cartesian Plane) though with the effect of magnetism they may act significantly more LIKE circles. (though never truly are)
the problem is that we do math based on the concept of perfect objects. as we know, they don't really exist, so we must instead turn to the idea that math is simply approximations based on the degree of accuracy that you need. (after all, ~39 decimal places will give you a margin of error of less than the size of an atom of hydrogen. a pretty acceptable value for most.)
quantum mechanics will introduce you the idea, though the theory still leaves much to be discovered.
Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...)
No.... 10a = 10 * 0.999...
10 * 0.999 is only equal to 9.999..... if .999.... = 1
10 * 0.999... was transformed into 9.999..., without showing which axiom allows this.
That is, this 'proof' requires you to assume an identity 10 * 0.999.... - 0.999... = 9.999... - 0.999....
I personally suggest taking this as a reductio of 0.999... . Maybe 0.999... does = 1, if the semantics of recurring decimals so describes it, but that being the case, I see little reason to ever use 0.999... over 1, save to deliberately obfuscate.
Myu:
... not the numbers themselves. The real number system differs from rational numbers in that uncountably many of them do not have a repeating decimal expansion and have to be represented by an approximation. That's what the decimal expansion is, a representation based on an approximation. What makes the approximation a valid representation is that, for any value epsilon (usually thought of as a very small number), the approximation can get within epsilon of the number.
..., which is a series that converges to one. And 0.999... = 0 + 9/10 + 9/100 + 9/000 + ..., which is also a series that converges to one. Therefore they're equally valid series for representing the same limit, one.
In other words, an infinite decimal expansion is a series that converges to a limit (the real number itself). So 1.000... = 1 + 0/10 + 0/100 + 0/1000 +
My point is that this is true by the DEFINITION of a real number. It's axiomatic. You don't prove axioms, because there would be only one step to proof, to point out that it's an axiom.
I once told a boss, "Well, you have to remember that half of all people are below the median in intelligence." He got all indignant and said "You don't know that! You can't prove that!" This thread reminds me of that altercation. And my own explanation just now reminds me of the fireworks factory explosion at the beginning of that Naked Gun movie and Frank Drebben saying "Nothing to see here! Move along!" If your goal is to come up with the funniest response, the correct response misses the point.
The argument against the above mentioned proof arises from philosophy: Actual infinities cannot exist. For example, 0.999.... mangoes cannot exist (ask why?). But 0.99999999999 mangoes and 0.999 mangoes can. Q.E.D
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This is absolutely preposterous. Of course you can add 1/9 to 1/9. The answer is 2/9. You are failing to separate what can be added as floats in base 10 with what can be added at all. For example, there is no reason that we cannot represent 1/9 in base 9 as 0.1, then add 0.1 to 0.1 to get 0.2 in base 9. This can all be done without approximation using floating point arithmetic, just not in base 10.
The only reason infinity might come into play is because 1/9 has no finite representation in base 10. But this is not a problem. Consider the number 1/10. Would you say that you can multiply 1/10 by 10? Assuming that you would agree to this, consider what would happen if you performed this operation in binary? In base 2, 1/10 has no finite representation, so by your logic, we cannot multiply it by 10. In fact, given any finite decimal there is another base in which it has only an infinite representation, and hence by your logic we cannot multiply numbers by decimals ever.
You have imposed an artificial limitation on the basic concepts of arithmetic by limiting yourself to floats in base 10.
I also have a problem with you stating that you can't use the value of pi in math when you absolutely can. The inability to write down infinitely many digits (by which I mean write down more digits than any finite number of digits) does not preclude the fact that we can do math with pi. Math is in no way limited to calculations which can only be done with finite precision. Take this example from calculus: find the area under the curve 4/(1 + x^2) from x = 0 to x = 1. We can show that the antiderivative of 4/(1+x^2) is 4arctan(x), and so the answer is 4arctan(1) - 4arctan(0). By the definition of the tangent function, tan(pi/4) = 1 and tan(0) = 0, so 4arctan(1) - 4arctan(0) = pi. That is, the area described above is exactly pi, or exactly the same as the area of half a unit disk (even though these regions look nothing alike). Notice that the answer is not that these are pretty close, they are in fact so close that given a set of tools with any arbitrarily small margin of error, we could not tell them apart. This is just one example of doing math with pi and not a definable approximation of pi.
Well, that proof abuses the loosly defined meaning of "equals". You cannot "sum an infinite series", though you can use that expression as shorthand for what's really going on. The sum converges on 1 as the number of terms approaches infinity, but that's not really the same kind of equality as 1 + 1 = 2. Other proofs are better.
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The problem with this approach is that you can no longer even define 0.999... The definition of this number is that it is the limit of the sequence sum_{i=1}^{i=k} 9/(10^k) as k "goes to infinity". If you have concluded that there is no infinity (in fact, I would agree with you here - infinity is not a number but a term meaning roughly larger than any number), and that infinity cannot be used in mathematics (with which I would disagree), then the number 0.999... cannot even exist.
So which is it? Either we can consider the concept of infinity, in which case 0.999... = 1, or we cannot, in which case the expression 0.999... has no meaning.
There most certainly are things "after infinity", the infinity you know about is actually the smallest infinity.
The cardinality of the natural numbers is the infinity called aleph sub null, it *smallest* infinite ordinal. The cardinality of all real numbers is aleph sub one, which is a bigger infinite ordinal.
http://en.wikipedia.org/wiki/Aleph_number
It isn't "far down" the line. No matter how far down the line you go, you'll never find any difference between 0.999... and 1. Therefore, they're equal.
Infinity is not merely a very big number. It really is just that: infinite. That's a far more important thing people have trouble comprehending.
Forget magic. Any technology distinguishable from divine power is insufficiently advanced.
Let me turn the situation around and see if it you see my point. The question asked was how long will it take it to come to a complete stop? If that ball bounced an infinite number of times, that means it bounced without end, it never reached a complete stop.
My point isn't that you can't calculate that the ball will stop bouncing in 30 seconds (or whatever), my point is that it will not have bounced an infinite amount of times before it stopped. Those two statements are diametrically opposed, if it bounces without end (infinite number of bounces) it never stops. If it stopped, it could not have bounced without end.
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The aliens are attempting control of our women -- fortunately this one managed to resist before they activated her laser eyes.
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I find this one to be the most rigorous.
I don't believe the problem is 0.999... It's how the hell to represent things like the result from 1/3. In computing at least, this kind of thing is a common cause of precision loss.
In whole numbers there is nothing between 1 and 2. So they are equal.
but symbolic computations are just that: symbolic.
What exactly is non-mathematic about symbolic computations?
you cannot represent much of out mathematics in the real world, which is where computation plays so wonderfully. give me a real world example of sin(pi)=0
What is the definition of sin? An elementary definition would be to say that the sine of an angle is the y-coordinate of the intersection point of a ray having that angle with the x-axis and the unit circle. We define pi as the circumference of the circle divided by the diameter, so in this instance, a ray from the origin having angle pi with the x-axis intersects the circle exactly halfway around, and as such has y-coordinate 0. That is, sin(pi) = 0. I don't understand what you mean by "real world," though. These are the definitions* - there is no other way to pursue mathematics than through precise definition.
If you, however, consider the definition to be based on a circle found in nature, then that is hardly a definition at all. You've already included in your definition all of the inaccuracy that we need (along with the ambiguity of deciding which circle found in nature should be used). This is why mathematics is not based on natural calculations, but is based in abstract logic and applied to calculations.
* - There are alternative definitions that could be used for sine, such as representing sine as a Taylor series. This definition can be shown to be equivalent to the one I've presented.
I can prove you, that the last binary digit of is 1, because if it was 0, we would just throw it out like in 0.110110 -> 0.11011.
There's no such thing as infinite smallness. I said that to highlight the absurdity of thinking that the number goes on infinitely while at the same time thinking it eventually ends. The entire point of an infinity is that it doesn't end. There is no space by definition.
You've dealt with such numbers many times without any problems. For example, in primary school when you learned about fractions:
2/3 + 1/3 = 1.
Or in decimal:
0.666... + 0.333... = 1
If you now find yourself wanting the revise the curriculum, you have a problem understanding the notion of infinity. :)
Well, that proof abuses the loosly defined meaning of "equals". You cannot "sum an infinite series", though you can use that expression as shorthand for what's really going on. The sum converges on 1 as the number of terms approaches infinity, but that's not really the same kind of equality as 1 + 1 = 2. Other proofs are better.
They're better in that they don't resort to limits, which some people haven't learned or have conceptual problems with.
But it's a perfectly fine proof. It uses the extremely well defined meaning of equals and the mathematically proven formula for calculating the sum of an infinite series, which it is just as possible to do as to say 9.99... which is infinite, or for that matter to say that the area under the curve x^2 from 0 to a is equal to (1/3)a^3. It's a very precise form of equals, not loose in the slightest.
The enemies of Democracy are
Oh, infinity gets more fun than that.
My favorite misadventure at the edge of rationality goes like this:
Suppose you have two containers. Each contains an infinite number of ping pong balls (okay, so they're containers of holding or some such) which are all individually number (1, 2, 3 and so on). No suppose you take a minute to sort out these balls. From one container you take them out in a nice, orderly manner (going 1, 2, 3) while in the other you just pull them out from the top. Now, we're dealing with a lot of ping pong balls here, so you have to be pretty quick about it. So first off get another couple of containers to put the with drawn ping pong balls in. Now get ready. In half the time (30 seconds) you have remove the first ping pong ball. In half the time left over after that remove another ping pong ball. Continue until your minute is up. Look what you've done!
The container where you tackled the job in an ordered manner is empty! The other container - despite having had the exact same number of balls removed - is full. Furthermore both of the other containers are also full.
At which point it's time to go grab a beer or other mind altering beverage, because things just don't make sense any more. And your arms are tired from lifting all those ping pong balls...
Every time I read a post about 0.999... my brain pauses after the number, allowing for omitted content. Which I guess there is... 0.000...001.
"...And who wants to make buttprints in the sands of time?" ~Bob Moawad
Is 0.888888888888888.... = 0.999999999999999999999999....?
If so, does 0.1111111111111..... = 0? or 0.1?
Or does 0.8888888888.... + 0.1111111111111.......... = 0.999999999999999999.........?
Whaaaaaaaaaaaaaaaaat. How does that work?
"16MB (fuck off, MiB fascists)" - The Mighty Buzzard
All I can think of is that it has something to do with the difference between countably and uncountably infinite. Even so... damn.
"16MB (fuck off, MiB fascists)" - The Mighty Buzzard
All geeks know that you can't write down one tenth in binary notation .... why is everybody so surprised when you can't write down one ninth in decimal? It's exactly the same problem.
The number '0.99999....' is only an approximation to the value "nine times one ninth", limited by decimal notation.
If you write down the same number in a more suitable number base nine the 'problem' completely vanishes.
No sig today...
I said "not uncommon", which doesn't mean the same thing as "universally true".
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How can you multiply .999... by anything at all? If the sequence is infinity, then any application of task or step can never be completed as it would take infinite time to perform the calculation.
Well you don't do it by long multiplication, because yeah you'd never get done. Instead you have to use other known and proven properties of multiplication to get the right answer. In this case the answer is especially easy to arrive at because multiplying by 10 in base 10 simply means you move the decimal place over one, and since there were infinite digits after the decimal, that doesn't change.
So 10 * 0.999... = 9.999... and you can do it in one step. :)
The enemies of Democracy are
Again, technically, and not perceptively.
Think of it like this, Quantum mechanics has a LOT of very interesting unexplored/unanswered areas, yet people still obsess over a damn hypothetical cat.
A patriot must always be ready to defend his country against his government. -edward abbey
My math skills aren't great, but is 10 * 0.999 really 9.999? Shouldn't it be 9.99? Where did the extra 9 come from?
I remember back when I was a graduate student reading about an alternate real number line where there existed a new number called "delta".
It was defined as being smaller than any positive real number and bigger than zero. Of course, this was not our normal real numbers that come from closing the rational numbers (using classes of Cauchy sequences) under the standard metric.
In these real numbers, 1/3 and .333... were not the same number, but were considered sufficiently close to be presented as the same answer to real world problems.
The advantage of this real number system is that it did interesting things to Calculus. All the complicated Epsilon & Delta limit theorems were trivialized and a lot of operations became simple algebraic manipulations. Also, things like integrals being the reverse of derivatives had interesting simplified proofs.
I also remember an argument being made that one could argue that this approach is not so far from reality. The reality is that in most cases we don't need more than 10 to 20 digits of precision. If we treated 10 to negative 80 as being this "delta" or essentially the same thing as zero (but not zero for calculus), you will find that mathematics does not fall apart as quickly as you might think and can still be essentially manipulated to give you most of the theorems and proofs of results critical to real world manipulations (including such things as General Relativity). And in fact, a lot of proofs become easier. This is not such a surprise to Physicists because they have been short cutting some of these types of proofs from the very beginning (starting we Newton who really did not quite grasp limits).
0.999... used in an equation is not actual inifinity, it is potential infinity.
0.999... - for which I personally rather prefer the notation 0.(9) - is not "an infinity". It is a perfectly finite number which happens to be represented by an infinite number of digits in its decimal representation. The ellipsis (or parentheses) represent that infinite number of digits, and nothing else. There's nothing "actual" or "potential" about it. There's no limit, there's no series. It's just a number.
GP obviously assumed that a and b (and x) are taken from the set of all real numbers, which is contiguous (and which is precisely why it even works).
I'm fairly sure you've only rephrased the problem. People who believe that 0.333... is only approximately equal to 1/3 will most likely have equal or greater difficulty believing that the sum of an infinite geometric series is exactly (initial term)/(1-(common ratio)), as opposed to a mere approximation. I don't think the equivalence between 0.333... and 3/10+3/100+3/1000+..., basic place-value representation, is really the source of the block—the problem is the idea that any infinite series can have an exact, finite sum.
Incidentally, this also affects the alternate proof given in the summary, since the same people would say that 10*0.999... (9.99...) is only approximately equal to 9.999..., since, after all, the latter form has one more "9" at the end. If "infinity" were just a very large number—which is how most people think of it—then they would be right, but that isn't the case. Infinity isn't a number at all.
"The state is that great fiction by which everyone tries to live at the expense of everyone else." - Bastiat
How can you multiply .999... by anything at all? If the sequence is infinity, then any application of task or step can never be completed as it would take infinite time to perform the calculation.
Same way as you calculate the limit of any other infinite series.
If .999... = 1 then
.999... - .111... = 1 - .111...
.888... != .999...
.999... != 1
and
so
So it is NOT true.
No sigs in BETA. Beta SUCKS.
You yourself wrote that it is the same number. A single number cannot be rational and non-rational at the same time.
(Note that "rational" is a property of a number itself, not a property of its representation.)
You're ignoring the possibility that some *irrational* number exists between 0.9999... and 1. In general any number of irrational numbers exist between any two rational numbers, even if there isn't enough space for "a single mosquito fart".
The whole point is that 0.(9) and 1 are not "two rational numbers". They are a single number.
So long as we're taking two different rational numbers, there is an infinite number of other rational numbers between the two. GPs point was that, if you could show any such number between 0.(9) and 1 (and prove that it really is between!), then you would disprove that 0.(9)=1.
What do you mean it's not a "real" number.
He means your syntax is meaningless. "..." means infinitely repeating. You can't say 0.0...1 because you've already specified infinitely repeating 0s. You could use some alternative syntax to specify some finite number of 0s followed by a 1, or a finite number of 0s followed by an infinite number of 1s, but you can't say "infinity... plus 1!" The concept is meaningless.
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There most certainly are things "after infinity", the infinity you know about is actually the smallest infinity.
Yes, yes, but you can't get from one to the other additively. "Infinity plus one" is not any bigger than infinity regardless of what kind of infinity you're talking about. The set of Positive Integers is the same size of infinity as the Non-Negative Integers. The first is infinite, you add the element 0, the result is the same kind of infinite.
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" Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999"
No. 10 times 0.999 does NOT equal 9.999, it equals 9.990. Kind of a big deal, really.
"The bigger the lie, the more they believe." - Det. Bunk
See Aristotle: http://en.wikipedia.org/wiki/Potential_infinity
The problem is not whether 0.999... equals 1, it's whether 0.999... is actually a well-defined.
All of calculus is built upon examining the importance of the *rate* at which lim x->0 1-x is creating 0.999...
When I look at lim (dy->0,dx->0) dy/dx , even though 1-dx and 1-dy are both 0.999... if calculated individually, the relative rate at which these two dimensions are going there still has meaning, leading us to the concepts of derivatives and integrals.
That's not "after infinity", that's "larger than infinity" (crudely speaking).
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It is potential because it cannot be represented in reality. To be able to represent it in reality, and make it an actual inifinity you would have to write all the nines. In which case you would never finish the equation due to restrictions of nature.
It's just an artifact of the decimal system. Other bases have different infinite fractions; for example, 1/5 (0.1 decimal) is 0.0011(0011)... in binary. On the other hand, 1/3 is just 0.1 in base 3. Any fraction whose denominator contains just the prime factors of the base (2 and 5 in decimal) will terminate; any other fraction will be infinite. So 1/2, 1/4, 1/5, 1/8, 1/10, 1/16, 1/20 all terminate (in decimal), but 1/3, 1/6, 1/7, 1/9, 1/11, 1/13, 1/15, 1/17, 1/18, and 1/19 are all infinite. Similarly, in binary only powers of two will terminate (1/2, 1/4, 1/8, etc.).
In terms of long division, non-terminating numbers (like 1/7) never end up with a remainder equal to zero no matter how many digits you compute, so there are always non-zero digits left in the result. Numbers which terminate (1/4) eventually reach a zero remainder.
So far as I know there are no real-world implications.
"The state is that great fiction by which everyone tries to live at the expense of everyone else." - Bastiat
And the second hurdle: people find it hard to believe that you can do mathematics with "infinity" as a meaningful quantity.
That's because if you just say "infinity" it means nothing, and it's easily confusing. It is only meaningful within context. Most will understand it when they have context. But you also have to keep in mind infinite sets have different cardinalities - that is where you lose most people.
Also a terminology issue, the number of ones in 0.111111... is indeed "Countable".
You can represent that number "in reality" (since it's just 1). You cannot write down its representation in decimal, yes, but representation of a number is not a physical thing anyway, it's an abstract concept.
I mean, you sure can use pi in equations, and then go on and solve them, despite the fact that pi does not have a finite representation in any base.
very well stated. I'd mod you up if I could.
the definition of a sine is nothing more than a ratio between the angle of a triangle and the length of the sides. in the real world, we cannot use undefinable units of measure to determine anything useful: we need to measure the degree of accuracy.
an example of this is the value of pi: how many times must you rotate a circle to allow any one point on the shape to return to the same point on a Cartesian plot? mathematically the circle must rotate [pi] times for this to happen thus sin(pi) = 0. but how do you determine when this rotation has completed? you can only do so by assuming a degree of accuracy: your point cannot be a "point" (they don't exist in the real world, as defined under the uncertainty principle) it must be an area containing the point you intended. there's no way to determine where that point is once measured the first time (under standard quantum mechanics) instead you can only estimate where it will be the second time.
in exactly that case, the point will NOT require a [pi] rotation to return to the original coordinates: as that would have to happen in 0.0 seconds, instead it will have traveled during Ts to a new location, resulting in the value being [pi +/- the degree of accuracy]
the point I'm trying to make is simply that: when you define something that can't exist (like an infinity repeating fraction) and try to perform arithmetic on it, the answers you get have no real value: they're just hypothetical values that have no place in the real world.
the 0.999...
math only exists to answer questions. there's no reason playing with the extents of the system in ways like this: as they have no real world ramifications (except that it reveals areas that the theory may be flawed)
just because the definitions are what they are, is what allows this problem to be so misunderstood. if I go back a few hundred years, I can prove without a possible doubt (on paper) that the earth is flat. one day, the definitions will change, and this "common knowledge" will be seen as false.
tl;dr: 0.999... = 1 only if you accept definitions that may or may not be accurate, and being that they cannot describe the real world at any scale: I assume them to be inaccurate.
Here is an article on decimal representation. By definition, it's equivalent to saying r = a0.a1a2a3a4... (all the digits). Every digit is placed at a specific location that can be identified with an integer index n, i.e., a(n). There is no upper bound to what the index n can be (that's what the infinity symbol is defined to mean here).
So I challenge you to identify the integer index (n) at which the proposed "1" is placed.
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10^(-infinity) is 0. It may seem like it could be 0.0...1, but what you really get is 0.0... and you never reach the 1 because there are infinite zeroes. Or think of it this way: 10^(-infinity) is 1/(10^infinity). 10^infinity is infinity. 1/infinity = 0.
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It's a limitation of the decimal system, not my understanding of infinity. Saying 0.999... is the same as 1 and believing it is true, highlights this limitation. If someone meant 1 they would have said 1 not bother implying the number closest to one without being one which is what 0.999... is.
Also, I mentioned this somewhere else, 1/3 does not equal 0.333..., it approximately equals it. Again highlighting the limitations of the decimal system. This is why you always do you calculations in fractions and compute the decimal result once at the end so the error in this approximation doesn't multiply itself.
The crux of the problem is you're really thinking about it all wrong when you don't understand it. The explanation becomes readily apparent if you start with a = .99, because then you actually have to do the math instead of just "lopping off" .999... off the end of the equation. When you do this, you realize when you lopped off the .999... you weren't actually subtracting, you were just dropping a symbol or placeholder for an amount of digits that can't be written off the end of the number.
.999... oh that's right, I didn't subtract .999... because that's impossible.
IOW
a=.99
10a=9.9
9a=8.91
a=.99
but wait how come when I subtract
but drawn to its logical conclusion, it becomes a=x=b, since there will be no number y between a and x or number z between x and b. It's ludicrous to think that for one number to be greater than another number there must exist a third number in the middle.
Here is an article on decimal representation. 1 has another infinite decimal expansion, i.e., 1 = 1.000.... 1/3 also has an infinite decimal expansion, i.e., 0.3333... So perhaps you would conclude that no multiplication can be accomplished on these numbers, either.
The solution to your paradox is that there are every-so-slightly more sophisticated mathematical techniques for getting multiplication done than mechanically multiplying each digit (for example, if you know that all subsequent digits are a particular fixed digit or a repeating pattern).
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No, because in both cases they are discreet balls, and both have the same ordinality. You can create a 1:1 mapping from the unordered set to the ordered set, so they are the same degree of infinity. So, doesn't make any sense to me. If it's right, I don't know why.
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but drawn to its logical conclusion, it becomes a=x=b, since there will be no number y between a and x or number z between x and b.
I don't understand where you're getting this from. He asked for any one number X such that A < X < B. He didn't ask for precisely one such number.
It's ludicrous to think that for one number to be greater than another number there must exist a third number in the middle.
Not at all - it's a perfectly valid claim so long as we're talking about the set of real numbers. This immediately follows from the fact that real numbers can be represented as points on a line - between any two different points, there is an infinite number of other points.
You acknowledge that "you can never get to the end of an infinite number", then make your case against the proof based on getting to the end and placing a zero there. That is FAIL.
The thing is, people who have difficulty equating 0.999... to 1 are specifically skeptical of this bit!
Also, it's really not true that "the sum of an infinite series" equals anything. It's a loose way of describing something in English that oversimplifies to the point of being wrong. It's a fine shorthand between two people who really know what's meant, but if you don't have that frame of reference the normal English meaning isn't right.
"Converges on" doesn't mean the same as "equals". Sure, "the number the series converges on" strictly equals 1, but if you skip over that bit for the sake of brevity in conversation, you've left out somehting important. It's kind of like not keeping track of terms you've divided by while doing algebra, with little nots of what can't = 0: it's fine to skip informally, but it matters formally.
And for this specific case, the idea that 0.999.. converges on 1 isn't debated, so proving that it converges on 1 misses the point of the argument. The skeptic says "I agree that it converges on 1, but that's not the same as equals 1", so to reply with a proof that it converges on 1 is a bit silly.
The right argument is: in the system of numbers we usually work with, 0.999... is just another way of writing 1 by definition. There are other, less familiar systems in which they are different by definition. You can take that conversation to interesting places, and teach more math, philosopy, and computability theory. Proofs within the system just waste everyone's time.
Socialism: a lie told by totalitarians and believed by fools.
"but .999... is not a rational number, it's a real number."
Also, Snowball is not a cat, she's a mammal.
Also, as my mother said, "We're not Methodists, we're Christians".
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As usual math people notoriously bad with language.
How long until it comes to absolute STOP ?
Infinity of time, because in the abstract universe the ball is bouncing in, distance is meaningless, and you could say that each subsequent bounce takes exactly the same amount of time as before, and to the limits of the problem, redefine each bounce as same height as before (this part may be tricky to you, but dealing with infinity can not be done in your standard your euclidean space, you need some trivial modifications).
If you mangle wording of the problem however, as mathematicians are quick to do (oh so good they are not in programming), what you say starts making sense to you.
This is not an attack, but an observation.
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You must be using some other form of logic that I am not aware of. You are talking about a bouncing ball, that is a physical object. If it is going to bounce an infinite number of times, it will take an infinite amount of time because it will never stop, ever. Hence, infinite amount of bounces.
Yeah, correct logic. ;) But don't feel bad, it is confusing, especially when there are multiple infinities at work.
The important thing to note in this case is that yes the number of bounces is infinite, but the time each bounce takes becomes infinitely small. At the limit, the ball will be bouncing infinitely small bounces infinitely fast. You can accomplish an infinite number of things that take an infinitely small amount of time in no time at all. :)
Basically, your statement is equivalent to saying that the sum of any infinite series (in this case, the series is of the time it takes for each bounce, and you sum them up to get the total amount of time it takes) must be infinite. But this is not the case; there are many infinite series which sum to finite numbers. 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... is such an example. It sums to exactly 2, because yes you're adding infinite things, but at each step they get closer and closer (infinitely so) to 0.
Another fun example of infinities working oddly together is Gabriel's Horn. Take the curve 1/x for x > 0, and rotate it around the x-axis to create a 3D "horn" shape. The mouth of the horn is infinitely wide, and the tail of the horn is infinitely long. The surface area of this horn is infinite. However, because the mouth becomes infinitely flat, and the tail approaches infinitely narrow, it turns out that the Horn actually has finite volume. Thus the tongue-in-cheek observation that the only way to paint Gabriel's Horn is to fill it with paint. :)
Obviously in reality -- as if you could even have a Gabriel's Horn in reality -- it would take an infinite amount of time for the paint you pour into the horn to reach the bottom. By the same token, in reality a ball would not bounce infinitely because it would not be a perfectly inelastic collision and energy would be lost
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I have no problem whatsoever with 1/3 = 0.3333...
I do, as they are not equal. one is the approximation of the other. The decimal system can't handle thirds and other fractions very well, it has to cheat. Either by using imaginary numbers with infinite amounts of decimal places, or rounding off to a nearby real number. This is fact. What is being done here is confusing people with a hack that manipulates/corrects a fundamental flaw in using decimal numbers.
It is only because decimal notation cannot truly represent 1/9, that we must allow ourselves to believe that 0.999... = 1 even though it is not absolutely true. This is because we believe 1/9 = 0.111... or 1/3 = 0.333... when they are not. Those are only approximations and thus using them in place of their fractions will result in a close but not absolutely correct answer. I will agree that in some instances 0.999... can be equal to 1, but not in all instances. In some instances it represents the number closest to 1 without being 1.
How can you multiply pi by anything at all? After all, not only does it have an infinite number of digits, we don't even know what they all are! How can you multiply 1/3 by anything at all? 1/3 = .333...
How can you multiply 1 by anything at all? After all, 1 = 1.000000... which is an infinite number of 0s.
Also, it's really not true that "the sum of an infinite series" equals anything. It's a loose way of describing something in English that oversimplifies to the point of being wrong. It's a fine shorthand between two people who really know what's meant, but if you don't have that frame of reference the normal English meaning isn't right.
That's actually kinda backwards. The normal English meaning doesn't seem right if you don't have the right mathematical frame of reference, and it seems as if we're talking loosely and informally about a different kind of equals. But if you have the correct mathematical background, then you realize we are using the term "equal" precisely, logically, and completely correctly, as is required for mathematical proofs.
"Converges on" doesn't mean the same as "equals".
But it does in cases where the series is absolutely convergent. That the sum of such an infinite series is equal to the number the sum converges on in the limit is a mathematical truth. That's the basis for the Fundamental Theorem of Calculus. It's the most important step in the proof of Euler's Formula that e^(i*x) = cos(x) + i sin(x). That the e, sin and cos can each be replaced with their equivalent Taylor Series and maintain the equivalence of the Formula is precisely what mathematicians mean by "equal".
The sum of the Taylor Series for cos(x) equals cos(x). 0.999... equals 1. These are both formally, precisely, and absolutely true statements.
The skeptic says "I agree that it converges on 1, but that's not the same as equals 1", so to reply with a proof that it converges on 1 is a bit silly.
A skeptic could incorrectly take issue with any step of any proof. I fully agree(d) that there are better proofs for explaining the concept to said skeptic, but it still remains true that the proof given is perfectly correct.
The right argument is: in the system of numbers we usually work with, 0.999... is just another way of writing 1 by definition.
No, not by definition. Our number system is not so sloppy that we have to rely on definition for such things.
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You're ignoring the possibility that some *irrational* number exists between 0.9999... and 1. In general any number of irrational numbers exist between any two rational numbers,...
Yes, but in this case we're not talking about two rational numbers, but rather a single rational number (9/9 = 1) written two different ways. Even irrational numbers must differ from each other (and other real numbers) by a non-zero amount, and the difference between 0.999... and 1 is exactly zero.
Consider this: 0.999... is greater than any other number less than one which you could possibly choose. Any two distinct real numbers must have infinitely many other real numbers in between, but there can be no number, rational or irrational, which is both greater than 0.999... and less than one. That means they must not be distinct, i.e. they must be the same number.
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How can you multiply pi by anything at all? After all, not only does it have an infinite number of digits, we don't even know what they all are!
Well, we don't. We take it to a particular digit then round off because a computer would endlessly calculate it.
How can you multiply 1/3 by anything at all? 1/3 = .333...
Again, we round it at some point based on the level of accuracy we require. .3333333 x 3 =/= 1
How can you multiply 1 by anything at all? After all, 1 = 1.000000... which is an infinite number of 0s.
Bad example. Infinite .000... still equals 0.
The point is that the proof is faulty in that the technique of multiplying both sides by 10 is assumed completed when, by virtue of being an infinite number, it simply can't. A completed operation of multiplying .999... by 10 assumes that at some point the operation ended at some unspecified point. You can't even plug this into a computer because someone would have to sit there for all eternity holding down the 9 key.
It's simply the fact that choosing a numbering system that is based on tens cannot precisely represent every possible number we want to represent. So we use symbols for those special somethings like pi, the square root of negative one, and we have clumsy but workable notation for things that repeat infinitely like 1/3.
We use portions of ten because we have ten fingers. If we had six fingers, we'd have no problem with thirds (2 + 2 + 2 = 10) but expressing five sixths in decimal would totally fuck with our finite heads.
Having said that, for all finite numbers, decimal is very damned good, but we should not pretend that it can accurately represent every value. Some numbers can be expressed in decimal if we introduce infinite ellipses. 1/3 is 0.3..., 3/3 is 0.9... or 1. We cannot express some numbers, like pi, because (as far as we know) they never happen to repeat. They are not neatly divisible into tidy little tens or tenths. Therefore we have symbols to represent them, and you replace the symbol with an appropriately-precise value for any real-world applications.
We have enough digits of pi to perform any calculation we as a species are likely to need to perform. If we need more, we'll crunch them out. We're OK with an approximation of it (hell, most people are OK with 3.14 as a value, some calculations like estimating paint coverage are perfectly OK with "about three and a quarter", some fast estimates are acceptable with "between three and four". Just a few more digits could calculate the circumference of the Earth to within a few feet, so we're pretty good, and a few more and we're into "distance to other solar systems within a few millimeters" territory). People who need more precise approximations of it use them. But few people pretend that pi is truly represented by any number they enter into their calculators. It's just "good enough that I bought enough paint without buying too much", "good enough so my building won't fall down" or "good enough to achieve orbit."
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How long until it comes to absolute STOP ?
Infinity of time, because in the abstract universe the ball is bouncing in, distance is meaningless, and you could say that each subsequent bounce takes exactly the same amount of time as before, and to the limits of the problem, redefine each bounce as same height as before (this part may be tricky to you, but dealing with infinity can not be done in your standard your euclidean space, you need some trivial modifications).
Meh. Yes it takes infinite time, in this coordinate space where you've defined the time and distance to be the same for each successively smaller bounce. You're basically just saying that it takes an infinite number of seconds to come to a stop if the length of a second is also asymptotically approaching zero.
In normal space-time (which works just fine, thanks), what happens is that each successive bounce is shorter in both distance and duration. The series consisting of the time length of each bounce converges, i.e. its sum is not infinite. It does not take infinite time.
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a = 0.999...
10a = 9.999...
10a — a = 9.999... — a
9.000...1a != 9
It's useful to imagine it does, but it remains an abstraction only. For all... physical purposes, it makes no sense.
That's the impression I've got anyway, you're invited to bash.
decimal would totally fuck with our finite heads.
I think you mean...uh...heximal? Is that the word?
i'd hit it so hard, if you pulled me out you'd be the king of britain [bash.org]
Yes, but in this case we're not talking about two rational numbers, but rather a single rational number (9/9 = 1) written two different ways.
I disagree. Let me make clear though that I think that "0.999... = 1" is the only reasonable and consistent interpretation of "0.999...". What I disagree with is that we're talking about numbers at all. I think we're talking about *notation*.
Is this nitpicking? Yes! It is! I'm not ashamed to nitpick when it comes to proofs. Why does it matter? Well, because not understanding that we're talking about notation leads to "proofs" that 0.9999... = 1 that are just as incoherent as the supposed "disproofs".
Take your argument. You make the assumption you that there can be no number between 0.999... and 1. That's fine, but you've pretty much *assumed* the conclusion by ruling out a Dedekind cut beween "0.999.." and "1", which the other side (if they understood more math) would disagree with.
The proof in the summary is *much* better, but it should start with several assumptions first that ensure that the subsequent operations are allowable. But that is *really* nitpicking.
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You were doing great until "Except 0.999... is not a rational number. It is an irrational number."
0.999... is as rational as 1 is, since they're equal.
Next time please don't post a story with 4 links. I don't know what to click!
Just because the U.S. is a republic does not mean it is not a democracy. Democracy/republic are not mutually exclusive.
When you multiply 5 by 5, do you add 5 to 0 five times or do you just put 5x5=25?
Same thing. You're working with symbols, not the actual numbers.
I didn't think this was new. My algebra teacher used this theory to prove it to us 15 years ago (yikes, I'm getting old).
Disagreeing with me does not mean you get to mod me troll.
Absolutely. No idea what the GP makes of e^(i*PI)=-1
Those two statements are diametrically opposed, if it bounces without end (infinite number of bounces) it never stops. If it stopped, it could not have bounced without end.
An infinite number of events does not necessarily mean an infinite amount of time. Figure the 1st bounce takes 1s, then the second 1/2s, then 1/4th, and so on... You get an infinite number of bounces in a finite amount of time.
Of course a ball bouncing infinitely is still impossible in real life, so let's look at something imminently possible: Your hand, waved through the air, passes through an infinite number of points, and yet arrives at the destination in a finite amount of time. How is it possible? The number of points your hand moves through is without end, so how can it ever finish? Well the answer is that there are an infinite amount of points, but only an infinitesimal amount of time is spent at each one.
This is, of course, ignoring the open question of whether the physical universe is continuous or discreet. I'm just saying, there's no mathematical issue with it being continuous, and the universe dealing with this infinity issue every time you move.
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What I disagree with is that we're talking about numbers at all. I think we're talking about *notation*.
You can talk about notation all you want, but when I say that 0.999... = 1 I am definitely referring to the equality of the numbers, not the way they're written. Notation only comes into play when it comes to accepting that the same number can be written multiple ways.
Take your argument. You make the assumption you that there can be no number between 0.999... and 1. That's fine, but you've pretty much *assumed* the conclusion by ruling out a Dedekind cut between "0.999.." and "1", which the other side (if they understood more math) would disagree with.
Perhaps it was badly worded, but that was not intended as an assumption, but rather a challenge to anyone who may disagree to come up with some number between 0.999... and 1. If such a "cut" is possible then an example shouldn't be difficult to find, but it quickly becomes rather obvious to anyone who tries it that (0.999... + x) > 1 for any real (and thus finite) value of x > 0.
The proof in the summary is *much* better...
I don't disagree—it's certainly more consise—but I do think it's much less persuasive to the non-initiate. Anyone who has a problem with 0.999... = 1, generally because they think 0.999... is "just an approximation", is going to have a problem with 10a - a = 9 where a = 0.999..., since that obviously isn't quite true for any finite approximation of a (10*0.9999999 - 0.9999999 = 9.999999 - 0.9999999 = 8.9999991 != 9).
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Correct. thanks. :)
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Notation only comes into play when it comes to accepting that the same number can be written multiple ways.
Yes, but that's *exactly* the sticking point. There is simply no argument that you can make that is psychologically convincing to somebody who hasn't grasped the distinction between the number itself and how we happen to write it down. It's particularly easy to get confused because the decimal numeral system is so helpful to computation.
Remembering that decimal number system is "notation" leads to clearer thinking on the problem. Of course a thing can have more than one name. Everybody knows that. That gets people over the hump that "the numbers look so different." They should be saying, "the ways of writing that number look so different."
If such a "cut" is possible then an example shouldn't be difficult to find,
Let's be clear here. I agree that a Dedekind cut is not possible if "0.999..." is a real number and the reals with addition and multiplication are a ring structure. However -- you can't reasonably claim that if such a cut existed, it would be easy to find. That's an appeal to intuition that happens to be right in this case, but that kind of intuition is horribly unreliable. Mathematics is full of numbers which are easy to describe by their unique properties, but hard to put your finger on.
Perhaps it was badly worded, but that was not intended as an assumption, but rather a challenge to anyone who may disagree to come up with some number between 0.999... and 1
The inability to do which proves nothing other than the person in question can't think of such a number. And when somebody is arguing the other side of this question, they're sure to grasp that.
The problem with your argument isn't the way it's worded; its the *idea*. Please don't take offense. You are right about such a cut being impossible (with the assumptions I've stipulated already). It's just that assuming that is for practical purposes assuming the conclusion.
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The whole point is that 0.(9) and 1 are not "two rational numbers". They are a single number.
Unfortunately, that's the very point in question.
I happen to agree that if
(a) 0.999... is a real number and
(b) we are talking about the real numbers as an algebraic ring, then
(C) 0.999... has to represent the same number as 1.
But you can't start off by assuming we're talking about only one number that if that is what you want to prove.
I was thinking about this in the car on the way home. You don't have this particular issue if you're an ancient Roman or Babylonian. It's a side effect of way decimal numbers aid calculation so wonderfully. Unfortunately, it's not perfect. The long division algorithm ends up generating these infinite digit sequences when really ought to spit out a simple rational number like 1 or 2/3. The only possible consistent thing to do is to consider the output of the algorithm as generating an alternative representation of that simple rational number. If you do, everything works fine.
I think this problem is a kind of linguistic bug. People confuse "numbers" with the strings of digits churned out by arithmetic, because of the decimal number system's amazing usefulness in computation. Unfortunately, the decimal representation isn't perfect. The division algorithm sometimes spits out infinite sequences of repeating digits because it doesn't have a natural notation for simple rationals like 1/3 and 2/3 (which added together are 3/3 or "0.999...").
That wouldn't be so confusing if we remembered that decimal is just *notation* for representing numbers. If we change to base 3, then dividing a number by three can be done with a simple decimal point shift. The very same calculation, with the very same *numbers* that produces an infinite sequence in decimal produces a nice string when we're in base 3.
So clearly, the infinite digit problem isn't a property of particular *numbers*. It's an issue of *notation*. If we think of it that way, then it's psychologically easier to accept that two different *representations* of a number could look different. After all things have multiple names all the time. As long as there is an infinite number of representations to work with, there's no problem with assigning any finite positive number of representations to each rational number (although admittedly that point is a bit subtle).
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That wouldn't be so confusing if we remembered that decimal is just *notation* for representing numbers. If we change to base 3, then dividing a number by three can be done with a simple decimal point shift. The very same calculation, with the very same *numbers* that produces an infinite sequence in decimal produces a nice string when we're in base 3.
So clearly, the infinite digit problem isn't a property of particular *numbers*. It's an issue of *notation*. If we think of it that way, then it's psychologically easier to accept that two different *representations* of a number could look different. After all things have multiple names all the time. As long as there is an infinite number of representations to work with, there's no problem with assigning any finite positive number of representations to each rational number (although admittedly that point is a bit subtle).
I agree that this is indeed the crux of the problem.
That said, changing representation does not always solve the issue with "infinite length numbers". For example, pi (or any other irrational) is infinitely long if written down in any positional system.
So strictly speaking they would be quantifiable in real numbers.
There is no such real number. The final 1 would be multiplied by 10^{-\infty}, which is either 0 or undefined as a real number. Hence, your "number" is either 1 or undefined.
From the summary:
Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1.
That is not a correct proof.
In order to perform the subtraction step in the given proof you had to let the thing after the decimal point be equal to a, but that is presuming your conclusion, which is a fallacy.
I.e., the thing after the decimal point in "10a = 9.999..." does not equal a. It equals 10a-9.
There are many simple, correct proofs, but that one is neither simple nor correct.
Here's a simple, correct one:
Define (0.999...) = {the limit as i goes to infinity of [1-(1/10^i)]}.
The right hand side is the same as {1 - [1/(the limit as i goes to infinity of 10^i)]}.
Now, the limit as i goes to infinity of 10^i is infinity, and 1/infinity is 0, therefore 1/(the limit as i goes to infinity of 10^i) is 0.
Therefore the right hand side is equal to 1. Thus,
0.999... = 1. QED.
Of course, this is only simple if you already understand the bog-simple concept of limits; but if you're smart, you'll use this as a simple example when teaching limits and thus prove that bob's your uncle.
But it does in cases where the series is absolutely convergent. That the sum of such an infinite series is equal to the number the sum converges on in the limit is a mathematical truth. .... The sum of the Taylor Series for cos(x) equals cos(x). 0.999... equals 1. These are both formally, precisely, and absolutely true statements.
You cannot take the sum of an infinite series. The "sum of the Taylor series" is a nonsense statement. You're either using shorthand for what's really true (which is fine if the long form is understood), or you're just flat wrong. (Much like pretending "infinity" is a number: handy shorthand, but formally nonsense.)
Formally, all you can say is "as n grows aribtrarily large, the sum of F(n) grows arbitrarily close to some unique x". The limit of the sum exactly equals x, by definition of the limit operator, which is needed precisely because you can't sum an infinite series. But it's the limit that equals something, not the sum that equals anything.
And that formal distiction makes this a poor choice of proofs to use, because a reasonable skeptic can still say "sure, the limit equals whatever, but that still doesn't prove that 0.999... equals 1, only that it grows arbitrarily closer to 1, which I have already stipulated."
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It has to do with infinity being a proper subset of infinity. The 1:1 mapping doesn't matter so much - technically you might empty the second container (if you're pulling ping pong balls out at random you could very well pull 1, 2, 3 and so on in order) but it's unlikely.
If you need further convincing - think about it this way. In the second container you take an ordered, but incomplete approach. In the first period you remove the ball numbered 10. In the second period you remove the ball 20 and so on. The end result is the same - infinite balls in the dump container and infinite balls in the start container.
To further confuse things, picture a third container. In this container we place an infinite number of balls and remove them one by one. But instead of starting at number one we start at number two. How many balls are left at the end of the madness? One.
You can accomplish an infinite number of things
In my understanding of infinity, you can never accomplish a infinite number of things, as you would continue having to do them, ad nausium, forever, hence infinity, the never ending series of events.
:).
I do appreciate your respectful debating of this subject, it is a pleasure to see it outside of Plato's The Republic, especially on Slastdot
In the end I think we are splitting hairs. I know to solve your given equation, you would solve for the limit as it approaches infinity, and you could argue exactly what you've argued and be correct, in theory. However, in reality, the ball would not have bounced an infinite number of times before stopping as a ball bouncing an infinite number of times can never stop. That situation is impossible (a ball bouncing an infinite number of times and then stopping) regardless of the amount of time it takes. It can bounce a whole mega-ton-ass-load of times, but not an infinite amount of times.
I will argue once again that in some cases 0.999... represents 1 (when rounding 1/3 or 1/9 to decimal and multiplying it by 3 or 9 respectively). however 0.999... also represents the number closest to 1 without being 1. Two valid theories of math conflicting. Since the first is an obvious correction on the limitations of precision of decimal numbers and the second is not, I believe the second to be more true.
The so call long tail have gone where? It just disappear? or it ever exists?
Add 5 to 0? No. 5x5 is a representation of 5 groups of 5.
.9 becomes 9. 0.09 becomes 0.9 - and an infinite sequence will never complete the mechanical action of moving forward without truncation. That's why the proof fails. This issue probably doesn't come up ever except in the situation where one tries to prove 0.9999...= 1. At any other instance, and for all other things we truncate or round an infinite sequence at the decimal place which is most convenient for accuracy.
An equation is a representation of mechanical action (albeit non-physical)where a number is the material in which we shape and adjust using various tools. (Showing your work in math is key, the complicated equation can be broken down into it's constituent portions, each with a specific mechanical action that it represents) Whenever we deal with an infinite sequence like pi or something else, we truncate the sequence in order to have an "end" to the mechanical action and then come up with a result based on that truncated result.
The problem with moving the decimal is that your not moving it back (from left to right) position, you're moving the entire sequence forward -
You cannot take the sum of an infinite series. The "sum of the Taylor series" is a nonsense statement. You're either using shorthand for what's really true (which is fine if the long form is understood), or you're just flat wrong. (Much like pretending "infinity" is a number: handy shorthand, but formally nonsense.)
Saying that the sum of the Taylor Series for sin(x) is equal to sin(x) is neither nonsense nor a shorthand for something that isn't precisely "equals" in a formal sense. It's is in fact formal mathematical truth, using the "=" sign as you'll see in any calculus book when it says "sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! +...". Not "the limit of the sum from 1 to n as n approaches infinity", but literally that infinite series is equivalent to the sine function. You can even do normal algebra, like dividing one side by x to say "sin(x)/x = 1 - (x^2)/3! +...", or grouping terms of this series with terms of another infinite series, like for cos(x), which is how things like Euler's Formula were proven. It's how calculus works.
I bolded what you said about infinity not being a number because it's highly relevant and you're 100% right. Infinity is not a number, and it's only used as one as a shorthand for situations where there actually is no number because of discontinuities or divergence. These are the situations where you can only speak of approaching the limit. 1/x at x=0 is not equal to infinity. However when there is an actual answer, like the sum of 1 + 1/2 + 1/4 + 1/8 + ..., which equals 2, the fact that it takes an infinite number of steps to get there isn't necessarily a problem, if you don't have to actually perform them to know what the answer would be. You don't have to actually add up infinitely many tiny strips under a curve every time to know that the area is equal to the anti-derivative plus a constant, even though that's how you prove that this is true.
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The ellipse after the decimal number (...) implies the completed infinite. It is a notation. The same way that 9/9 is a notation. Using this notation, 1/3 = 0.333... exactly. What these notations represent really are equal, and not an approximation that you claim. The problem a lot of people have with math is understanding the notation. Your post is wrong because you do not understand it. Until you understand the notation, it is understandable that you make the claim that you do. And in the set of real numbers, and even in the set of rational numbers, there is no number that is closest to 1 without being 1. Such a number does not exist. Point is, 0.99... = 1 in all instances, because that is how the notation is accepted by the math community. Don't like it? Make your own notation. I'm fine with it the way it is.
The sum of an infinite geometric series is equal to a/(1-r) which a equal the first value and r is the ratio between the values. .9999..... is equal to 0.9 + 0.09 + 0.009 + 0.0009 ....
The common ratio is 1/10 or 0.1. The first number is 0.9
a = .9 and r equals .1 .9/.9 == 1
This works for all infinite series with a common ratio where r greater than -1 and less than 1 .6 .1 .6/(1-.1) = 2/3
a =
r =
Infinite series are great.
To some degree, this is true. 0.999... does represent the number closest to 1. That is, if we consider the interval (0,1) (which is all of the real numbers x such that 1 > x > 0 and x is not 0 or 1), 0.999... represents a the limit of a convergent sequence of numbers (0.9, 0.99, 0.999, ...) within this interval. Your thought is that since each of these approximations for 0.999... is less than *but not equal to* 1, 0.999... must also be less than (and not equal to) 1.
Sadly, this is not the case. For proof, let a and b be numbers. If b - a = 0, then by simple arithmetic, a = b. Hence by the contrapositive statement, we have that if a != b, then b - a != 0. Without loss of generality, assume that b is larger than a, so that b - a > 0. We can agree that 1 is not less than 0.999..., so in particular, we have that 1 != 0.999... if and only if 1 - 0.999... > 0.
Now, we know that for each approximation a_n of 0.999... (a_1 = 0.9, a_2 = 0.99, a_3 = 0.999, etc) 1 - a_n = .000...01 (where there are n-1 0s). And since 0.999... > a_n for each n, 1 - a_n > 1 - 0.999... for each n. In particular, this means that 10^{-n} > 1 - 0.999... for any natural number n.
Suppose for the sake of contradiction that 1 != 0.999.... Then 1 - 0.999... > 0, say 1 - 0.999... = e. Consider the reciprocal 1 / e. Because e > 0, 1 / e is some (possibly very large) number. Because 1 / e is a real number, we can find some n such that 10^n > 1 / e (This is the crux of the argument, which is sometimes called Archimedes's Principle. Most people agree that no matter how large a number you choose, I can choose another one that is bigger). But then e > 10^{-n}, so that 1 - 0.999... = e > 10^{-n} for some natural number n. This is a contradiction, since we found above that 10^{-n> > 1 - 0.999... for every natural number n. Because of this contradiction, one of our assumptions must be wrong, and our only non-obvious assumption was that 1 != 0.999.... Therefore 1 = 0.999...
The problem with your argument is that there is no number closest to 1 without being one. Just as I can always choose a number bigger than a given number, for any number less than 1 there is another number between that and 1.
not additively, but by repeated multiplication - 2 ** aleph-zero = aleph-one
"Valid" in what sense? The 0.999... is merely notation; it's entirely up to our definitions what it means. The only deciding factor is what's useful. We declare 1=0.999... (or rather, declaring that both are in fact just convenient notation for a dedekind cut) because we declare we're using the "real numbers" (a ridiculous term, but there we go), which form a consistent structure which is effective in modelling reality (e.g. geometry, probability...). If you want to construct another system that's fine, but you'd better come up with a consistent notion of what "the number closest to 1 without being 1." actually means (and even then you should probably use a different notation for it, to avoid confusion).
I am trolling
You're wrong. But it'll do you more good to work out why you're wrong than for me to spend time walking you through it.
If you can't concede the possibility that you're wrong, it would be a waste of my time anyway.
Well, I'm not sure this argument can be settled, but I don't see how you can say that dropping the limit operator from math is anything but informal shorthand. Describing a series as a "sum from 0 to infinity" is fine shorthand, and I don't bother the write the limit operator either, but I know it's there. Writing the Taylor Series for sin the way you did is just like writing "sin(0)/0 = 1" - true at the limit, and I know what you mean informally, but formally you can't do that.
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Math with pi:
Area of circle A (radius = 1 unit) = pi*r^2 = pi square units
Area of circle B (radius = 2 units) = pi*r^2 = 4*pi square units
Radio of the area of circle B to the area of circle A = (4*pi) / pi = 4
No approximations required.
"The state is that great fiction by which everyone tries to live at the expense of everyone else." - Bastiat
You can create a 1:1 mapping from the unordered set to the ordered set...
Are you sure? How would you do that without imposing some kind of order on the unordered set? Simply choosing balls at random and assigning sequential numbers wouldn't work—for any natural number n, there are a finite number of balls numbered less than or equal to n, and an infinite number which are greater. That means (for any value of n) the odds of choosing a ball numbered less than or equal to n is zero, i.e. such balls will never be chosen, even after infinite choices. So even after removing the infinite set of balls numbered greater than n, there must always be another infinite set of balls left numbered less than or equal to n (with n --> infinity).
Basically, the ordered case removes every ball in the set (1...+inf), while the unordered case splits the infinite input set into two infinite subsets (1...n and n...+inf with n --> +inf) and only removes the second subset. Moreover, repeating the process would just keep splitting the remaining infinite subset into smaller (but still infinite) pieces.
"The state is that great fiction by which everyone tries to live at the expense of everyone else." - Bastiat
Well, I'm not sure this argument can be settled, but I don't see how you can say that dropping the limit operator from math is anything but informal shorthand.
Because in cases of convergence you can mathematically prove that the value of the limit is in fact the sum at the limit, and instead of talking only of a limit use the infinite sequence itself in place of the sum, and vice versa. If it doesn't converge, then the answer is infinity, and then you can only talk about approaching that infinity. However an infinite number of additions that sum up to 2, or cos(x), can be spoken of without the limit.
The way for this non-argument to be settled is for you to pick up a Calculus book so you can understand how this is actualy done, formally. Look up the Fundamental Theorem of Calculus, Taylor Series and other infinite series, Euler's Formula, and other proven aspects of mathematics that depend on convergent limits. Look how it is proven that e^(i*x) = cos(x) + i*sin(x). That thing you didn't like with how I wrote the Taylor Series? Integral to the proof, and I assure you Euler's proof was quite formal.
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The inability to do which proves nothing other than the person in question can't think of such a number.
Right, but I'm not looking for a proof. I realize that my argument isn't a proof as stated. That's fine. I'm just looking for a way to get the other person to accept that 0.999... and 1 are the same number. Assuming that they brought up (and accept) the Dedekind cut concept, trying and failing to come up with any number in between should do more to persuade them than the argument in the summary.
As for proof, if we start from the principle that every real number has at least one decimal representation (possibly infinite) which can be compared digit-wise with any other real number's decimal representation, then it follows that 0.999... must be greater than any number whose decimal representation starts with "0." followed by any sequence of digits containing one or more digits which is not "9". Any real number greater than zero and less than one which is not 0.999... must contain a digit which is not "9" (since otherwise they would be equal), so any such number must be less than 0.999..., and not between 0.999... and one. Q.E.D.
"The state is that great fiction by which everyone tries to live at the expense of everyone else." - Bastiat
It has to do with infinity being a proper subset of infinity. The 1:1 mapping doesn't matter so much - technically you might empty the second container (if you're pulling ping pong balls out at random you could very well pull 1, 2, 3 and so on in order) but it's unlikely.
I see, it makes sense looking at it from a set theory perspective. In one case, you're subtracting set of positive integers from the set of positive integers. In the other you're subtracting an infinitely large set of random positive integers from the set of positive integers.
The enemies of Democracy are
Using this notation, 1/3 = 0.333... exactly.
No, it does not. Decimal notation cannot represent 1/3 exactly, it can only attempt to do so with the irrational number you listed. This is a cheat, a hack, an exception to the rule. And because of this people accept that when they see 0.333... that it most likely is 1/3 because they understand the limitations of decimal notation. I cannot see anyone trying to use 0.333... in any other manner, so I have no real problem with this hack. However, someone could use 0.999... to represent the number closest to 1 without being 1 very logically (how else would you represent it). Yes, it is an irrational number, but so what? Irrational numbers are used all the time in math.
But my point is allowing people to be accustomed into thinking 0.999... = 1 when that notation can also be used to define the irrational number I mentioned is just bad form. It also only serves to confuse people, unnecessarily.
I’m too confused by the impossibilities in the analogy to even begin to see what you’re trying to say about infinity.
E.g. the two containers would have to be infinitely large, there is no possible way that you could sort them in a minute – hell, it’d take infinitely long just to find the ball numbered “1” in the one that you’re attacking in an orderly fashion – and going infinitely fast at the end, as you suggested, is impossible; furthermore, if your containers are infinite in size, how is “full” a useful adjective when describing them?
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
No; it is true, and in fact the distance traveled by the ball will also be finite and calculable, because each bounce is some constant percentage smaller than the previous.
Suppose the ball bounces to 1/2 its original height with each bounce, and you toss it upward from the ground with just enough velocity initially that it reaches a height of 0.25 meters. It travels 0.5 meters in total (0.25 meters up and 0.25 meters back down) before its first bounce, travels 0.25 meters on that bounce, 0.125 meters on the next, etc. How far does it travel in total before it stops bouncing?
The answer is 1 meter.
This image visualizes exactly that... and since this is Slashdot I will also point out that the distance it travels is the repeating decimal 0.11111... in binary.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Please tell me what irrational number I listed. Every number in my post is a rational number.
Most likely? I don't understand that. Are you saying the notation expresses a probability now?
Please tell me this number that is closest to 1 without being 1. I guarantee that whatever number you chose, I can find a number closer (all I have to do is take the average of the number you chose and 1). Thus, such a number does not exist. It is pointless to talk of something that does not exist!
No, 1/3, 8/9, 9/9, 0.9..., 0.3... and so on are not irrational numbers. They are rational. Numbers like pi and sqrt(2) are irrational. You don't understand what irrational numbers are, you're claiming you understand the notation when you don't. There is no cheat, and no hack, and the entire point of this entire thread/forum/story is to point this out. If this really doesn't make sense, I'm sorry. Otherwise, I've been trolled, so congratulations.
Nice cop-out. That position suggests you don't really know how to explain it and rather than admit it you want to be superior. Generally the only other type of people that take that position are successful yet egocentric people who already feel superior and feel that explaining things to the plebes is beneath them.
I'm not a mathematician, I said that early on. However I am well versed in behavioral linguistics.
if "1" = a concrete idea and "0.(9)" = an abstract idea then 1 0.(9) a concrete idea an abstract idea
Apparently it dropped some of my last comment. It was supposed to read:
If "1" = a "concrete idea" and "0.(9)" = "an abstract idea" then 1 0.(9)
a concrete idea is NOT an abstract idea
Please tell me this number that is closest to 1 without being
I did, 0.999... (repeating nines to an infinite number of decimal places) which is also why it is an irrational number.
No you didn't. You still don't understand what an irrational number is, or the notation. http://mathworld.wolfram.com/IrrationalNumber.html
Read some of the comments first and hopefully you'll understand.
If I used a sig over again, would anyone notice?
Fail. There is no such number 0.999...8. With an infinite number of 9s, you can never get to the end to tack on the 8. If you follow Lightstone and say you can, then, fine, you can prove all kinds of weird things. But your proofs are based on a false concept so they're not valid in my world.
If I used a sig over again, would anyone notice?
And I notice that your reply isn't an invitation to discuss further but rather a thinly veiled attack on me. Indicating my surmization of the situation is correct. Winners all round. I don't wast my time and you can go on thinking that mathematical symbols represent mechanistic operations.
Yes, you are. 10a - a ALWAYS equals 9a, not 3a. So you would have 9a = 3, which makes a = 3/9 and thus .3333... isn't as weird as .9999...
What trips people on the 0.9999... that they think that the number 1 is the unique representation of the value. What this proof says is that 0.9999... isn't a separate number at all, just a different representation of 1. Try to think of it more as a limit probably than an arithmetic if that helps: Each digit added to 0.9999... brings it close and closer to 1, just as ever increasing deltas for a limit bring the limit closer to a value. Take the pattern to infinity, at the limit literally is EQUAL to the value the deltas head to. It's a little hard to grasp infinities sometimes. The same is said of 0.333... Each digit of 3 added brings the value closer and closer to 1/3 until, at the limit, it is literally equal.
Or, if it helps, don't think of 0.999... as a separate number at all (it isn't) and just think of it as simplifying to an equivalent value, just as 3/9 simplifies to 1/3.
Article is talking of an infinite number of 9s after the decimal, which indeed does equal 1. You're correct that a limited number of 9s after a decimal does NOT equal 1.
You're making infinity into finite by saying that you can't "pull an extra 9 off the right side of infinity." By definition, there IS no "right side" of the number. It's infinite. You could multiply an 0.9999... by 10 and INFINITE number of times and it would still have an infinite number of 9s after the decimal and NO zeros whatsoever.
An infinite sequence multiplied by a number is still an infinite sequence. If you lost digits (or if you gained digits!) it would, by definition, not be infinite to begin with. Your logic is coincidently taking only a finite number of digits from infinite sequence, whereas the article retains the infinite sequence, as it should.
Yes, you CAN multiply 0.3333.... by 3 and get 0.99999.... Your little O(1), which is a sequence of 3s, also gets multiplied by 3, which is also a sequence of 9s...Go to infinite sequence math, it'll become clearer. And you can multiple 0.3333... by 3 and get 1. (That one's easy to see, 0.333... = 1/3).
The "proof" leaves a lot to be desired, but there ARE proofs that involve limits that do work much better. In that sense, in both the real and abstract world, yes, 0.999...=1. And no, there's never a 0.000...09 left over because then the number wouldn't be infinite in the first place. There's no "end" or "left over" to an infinite number. :)
You do realize that you incorrectly stated that .9995 is bigger than .999 repeating, right?
You realize that ".999 repeating" is the same as ".9999 repeating", which is clearly bigger than .9995.
I'm not sure there's a really good way to respond to this. I was just adding to the great grandparent's comment that infinity gets weird as hell with my own example of infinite weirdness (actually, the counter-intuative nature of infinity).
If, after reading my post, your first problem is that "it'd take infinitely long just to find the ball numbered "1" in the one that you’re attacking in an orderly fashion" then we're going to have a problem no matter where we go with the discussion. If only because you're real question should be "where the hell did you get infinitely many ping pong balls - let alone two lots of them and a container to hold them?". It's a thought problem rather than one of real world application - much like the great grandparent's thought problem about the ball.
As for using "full" as an adjective - I'll cop to that. It's easier to type that "contains infinitely many ping pong balls" and has a meaning a meaning similar enough for the purposes of my example. My apologies.
If only because you're real question should be "where the hell did you get infinitely many ping pong balls - let alone two lots of them and a container to hold them?"
Yes... well... that was my first question, but I was willing to let at least that much slide for the sake of the analogy.
Also, you did manage to have infinitely many times more apostrophes in that quoted sentence than you should have. :p
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Well, we don't. We take it to a particular digit then round off because a computer would endlessly calculate it.
No... I can use perfectly good algebra as follows, and obtain an exact (rational!) result without rounding:
Area circle 1, radius 10’ / Area circle 2, radius 20’
pi x 10^2 / pi x 20^2
10^2 / 20^2
100 / 400
1 / 4
I was multiplying by an irrational number with an infinite number of decimal digits... yet in the end we found that circle 1 is exactly 1/4th the area of circle 2 – no rounding or irrational numbers involved, because pi was simplified out of the equation.
In fact, I could generalize it and say that the formula for any two circles will be (r1 / r2)^2.
Infinite .000... still equals 0.
By the same logic infinite .999... equals 1. It isn’t an infinite sequence of digits that we need to calculate infinitely before we can know the result. It’s the concept that is expressed by them: the summation of the geometric series f(n) = 9/10^n, as n goes from 1 to infinity... calculus tells us that it converges to 1.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Add 5 to 0? No. 5x5 is a representation of 5 groups of 5.
And how many things are there in 5 groups of 5?
I think you missed his point... when you count 5 groups of 5, you start with zero, and add 5 to it 5 times. Or you do multiplication and conclude that, without counting, you know that there are 25 objects because that’s just how multiplication works.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Because in cases of convergence you can mathematically prove that the value of the limit is in fact the sum at the limit,
Not without adding an axiom you can't. The limit operator lets you say "even though we can't sum to infinity, if we could this is the only thing it could equal". The result of the limit operator is a normal value, but only because the limit operator lets you do that. You might try picking up a math textbook beyond intro to calculus ...
Euler used his own notation for many things, since he was so far ahead of everyone else, so it's not useful to reference him here.
Socialism: a lie told by totalitarians and believed by fools.
This wasn't Euler's "notation". It's the current proof as given in Calculus books for Euler's Formula, which is still a formally-correct equation using equals in a formal sense. Look it up, because you obviously have no idea what I'm talking about, or you're talking about for that matter.
FFS, I don't care that your understanding of how limits work means you think it must be "informal". You're wrong, and need to learn. According to you, the Fundamental Theorem of Calculus, which depends on taking smaller and smaller pieces of a function to the limit of infinitely small, is wrong to formally use "equals" in describing the relationship between a function's integral and its anti-derivative. Which means you're clueless about the basis of calculus (which I guessed because you repeatedly avoiding even addressing the issue of the FToC). So it's pretty hilarious to hear you saying I should try going beyond intro. I have, and this concept remains important and is never contradicted. You haven't, and need to go back to Calc I, where they'll teach you more about limits than what you learned in pre-calc.
The enemies of Democracy are
Nuh-uh, you're the stupid-head.
Socialism: a lie told by totalitarians and believed by fools.
It's true, I'm pretty stupid, and my head especially so. Maybe if I wasn't, I could overcome your resistance to education.
The enemies of Democracy are
Ohhhh well, considering the bartender never in theory approaches 2 beers if infinity of mathematicians order each successively halved portions, well... Maybe the fact that beers are easily quantizable makes it much easier to agree with a premise.
When you (I, rather) try to quantize an analog phenomena, my imagination, which is quite reasonable mind you, enters a sort of wondrous state of incapability of division -- but well, you should at least partly agree that analog space is hard to divide, even in abstraction.
This is why I think that reality is mostly digital, as in, quantizable. Because my mind operates in, and understands, discrete phenomena much better.
I AM BIASED :P
I have nothing to lose but my bindings.
That would be floor(r). Why do you think it is not uniquely defined?
One can be used for counting. The other cannot.
Well, one could count "1.0", "2.0", if they really wanted to. In any case, subsets having properties that their supersets do not have is hardly unusual.
E.g. in an OO computer language we may have:
Class Integer inherits from Real
function Count()
We see that we can call function Count on an Integer, even though all Integers are of class Real. Since every Integer is a Real, we can do everything we can do to a Real to an Integer (though the result might not be an Integer). However since some Reals are not Integers there are some things we can do to Integers that we cannot do to (all) Reals.