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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
They have mathematics in Montana?
No sooner do I get over one, then you put a better one right next to me. Bastards.
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
Uh wtf, I did this in the 8th grade.
I can't help but feel like proofs that 1 = .999... are along the same lines as that joke proof that tries to prove all numbers are the same. I know these proofs are much more legitimate, but my intuition, in the back of my mind, screams "bullshit!" despite me knowing better.
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.
"both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1." But a=.999999......... this is a bit contradictory. I think we need another variable. a!=1 if a already = .999.......
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
Seriously, this kind of bullshit elementary algebra is slashdot news?
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.
Why is it faulty, because the proof is obviously wrong. 9a can't be equal to 9 and 8.99999..991 at the same time.
First off, by multiplying by ten, they lost one 9 at the end of the series. thus if using 6 decimal points (a = 0.999999, 10a = 9.999990. 10a - a = 9.999990 - 0.999999, 9a = 8.999991).
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.
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
For some reason, discussion of 0.999... = 1 seems to bring out the worst in people on forums. That, the Monty Hall problem, and the question of whether or not a plane on a treadmill can take off seem to be the evil trio of simple, solvable problems that lead to flame wars and arguments as intense as those normally seen in more subjective realms (religion, politics, abortion, tipping, and favorite text editor).
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?
A cat has been chasing a mouse for months. He's never been able to quite get hold of it! Every time, the mouse either jumps into a hole, hides under a sofa, or skips off somewhere else just out of reach, and the mouse is fed up. The mouse says to the cat "Ok, I'm fed up. I don't want to be chased anymore. I'm tired and old, and the chase is too much." The cat is obviously elated! "I'll make you a deal, Mr Cat" says the mouse. "I will stop running and let you catch me, but only if you move half of the distance towards me with each step!"
The cat can't believe his luck! "Of course! Of course, I will only move half the distance with each step, as long as you do not run from me anymore!"
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...
TL;DR: 0.999... != 1. It's just really, really close. I don't care what number say; If you can show me an infinite accuracy measuring device, I'll show you a 0.999... unit of length structure is not 1 unit of length.
Finally had enough. Come see us over at https://soylentnews.org/
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.....
You are being very imprecise. The LIMIT expressed by the infinite series 0.999... is equal to 1.
But math is hard, and I don't expect you girls on /. to understand it.
I once saw this problem on a 9th grade algebra problem set. It was with a bunch of easier decimal to fraction conversions. I spent three hours on this one, constantly getting the same answer! I finally found the answer on a PHILOSOPHY discussion board!
If we let a=0.999, then wouldn't 10a = 9.990, not 9.999?
Makes a pretty big difference.
Long signatures suck.
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!
OK, for those of you repeating the same old boring 'proof' that 0.9999=1 by 1/9 *9=1 it comes down to this:
What do you MEAN by 0.999... ? Mathematically, what is this object? If it's the sum n=1 to infinity of 9*10^(-n) within real analysis then yes. If you're defining it to be 'the decimal representation of 1/9' then yes by tautology. But sadly that's not ALL it can be - it can be a member of the hyperreals, valued 1-w where w is the smallest number (read up on hyperreals before you respond please - http://en.wikipedia.org/wiki/Hyperreal_number would be a good place to start). In that case it is most definitely NOT 1. TFA has a good list of things that this symbol could mean, and a discussion of the implications of each. I'd advise you to read it and learn what the issues are before effectively making the DEFINITION that 0.999...=1 in order to prove that statement.
?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?
multiply by ten and you must have a zero at the end (on the corresponding decimal place of the last 9 of 0.99999, no matter how far down the line that last nine is)
therefore you're actually left with 10a - a = 9.9999999999...0 - 0.99999999999...9
9a = 8.999999999...1
a = 0.9999999
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).
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.
[
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.
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.
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.
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}
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!
Read it.
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.
.. simplify - forget all the fancy maths
Say your number was 9.999... (finite N decimal places)
mulitplying it by 10, SHIFTS these LEFT.
Leaving (in a finite) series that 1/nth spot as a Zero.
so subtracting say (.99999 from 9.9999)
does NOT result in a 9, but infact a 8.9991,
similarly for a LARGE number of decimal places - it would extend to 8.999......1.
Now extend this to infinity ..
The number obtained - would have to prove that a convergence between
8.9999.....1 at infinity and 9 existed.
So you havent proved 1=.999, you've just shifted the proof to the boundary of infinty .. DUMBASS !!
-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!
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.
It's infinate you see.
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"
nt
1 - infinity^(-1) = 0.999 ?
1^(-1) = 1 --> 1^(-1) > 0
2^(-1) = 0.5 --> 2^(-1) > 0
. . .
n^(-1) = m --> n^(-1) > 0
therefor
infinity^(-1) > 0
Let's take unbalanced ternary first: 0.22222222(repeating) = 1.0, the same form as in decimal.
In balanced ternary, this visual difference doesn't happen. Instead of going 0.2, 0.22, 0.222, etc, you go 1.1, 1.01, where 1 = -1. (Slashdot doesn't like underlines, and using (-1) looks clunky)
Thus, instead of .222 222 2..., you get 1.000 000 0...
With balanced decimal, the same thing would happen. Instead of 0.999 999 9... you would have 1.000 000 0....
As a bonus, balanced notation removes the incentive of stores of using .99's in prices.
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)
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."
.99999..... != 1/9
Repeating decimal notation does not represent a fraction. It's a limitation in the notation. Just because your math teachers in schools may have told you to be exact to the fifth digit doesn't mean that is how mathematics works.
Let a=0.999
10a is not 9.999, it's 9.990.
Seriously this is now worthy of publication?
E
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.
I lost you at "We know that "2x" is always greater than "x"".
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
3D may not be being adopted has quickly as TV manufactures would like, but I don't think it is going away. This isn't the same once every decade or so gimmick it was. But the camel's nose under the tent won't be movies I think, but rather immersive 3D games with good 3D tracking.
I haven't yet tried Sony's Move system, but couple 3D tracking with a large 3D display and you may have an unbeatable gaming experience. I am also not a Second Lifer or a WoW player, but again 3D seems ideal for when you are not just looking passively at a story being told, but must move about in an environment. 3D Desktops have been predicted for quite sometime, but perhaps you really need true 3D to pull of a 3D Desktop.
Still this may all fall to wayside if someone can get rid of the screen, giving you true mobility in a 3D space. Yes there are VR 3D headsets, but they are clunker than the 3D glasses everyone here is already complaining about and high definition VR headsets are prohibitively expensive. No doubt technology will eventually catch up with how to make a high definition, light weight, untethered, long battery life, unobtrusive, VR headset.
On a related note, more than 3D for passive content, we need higher frame rates. There seems to be some conception that movies must be in 24fps to have a 'movie' feel as opposed to a 'TV' feel. I don't know any TV in full progressive 60fps. Most prime time TV shows are shot on 24fps film. 60fps 1080p would be much more immersive for high motion scenes. Someone needs to shoot some action epic in 60fps or higher and see if the public responds to it. IMAX once sometime ago shot one or two films in 48fps. It was insanely expensive to pull off back then, but now should be a cinch. Oddly almost every one's HDTV is capable of displaying 60fps, but unless you are using it for gaming it probably never use more than half this bandwidth.
Letter To Iran
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?
Wait... So you're saying that someone who wrote a 28 page academic paper on the subject is wrong because they "didn't learn the rules of dealing with infinite sets"? Don't you think that the issue might be just a touch more complex than you're implying?
Yikes.
1. Repeated decimals are not irrational numbers EVER. 0.3 repeating for example is 1/3. 0.12345 repeating is 12345/99999.
2. Infinity is not an irrational number. I don't even know where you got that.
3. Yes, 2x > x makes no sense for x = infinity. It works just fine for irrational numbers though...
The DEFINITION of a decimal number 0.abc... is defined to be the infinite sum a/10 + b/10^2 + c/10^3 + ...
Two decimal numbers are DEFINED to be equal if these sums are exactly equal, NOT if their decimal representations are identical.
1.0000... has value 1 + 0/10 + 0/10^2 + 0/10^3 + ... = 1 as you'd expect ... = 1 too.
0.9999... has value 0 + 9/10 + 9/10^2 + 9/10^3 +
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.
[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?
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?
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
If you multiply by 10, you don't get 9.999. You get 9.99. Remove 'a' in your next step and you get 9a = 8.991. get rid of that 9 and you end up with a = .999
Go back to school
This isn't actually true. .999... represents the same number as the rational number 1, yes, but .999... is not a rational number, it's a real number.
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.
Well, ok: Let's use the theoretical concept of what an "asymptotic approach/asymptote" is technically -> http://en.wikipedia.org/wiki/Asymptote
Iirc/afaik, it means that if you try to say, reach your neighbor's doorstep from yours, but, only approaching 1/2 of the intermediate distance from your starting point to the door, & always cutting the distance left over in 1/2 everytime you try this?
Well - Technically/theoretically, you'll NEVER reach there, because you're only getting "1/2 way there" each time, but never ALL the way there.
So, that said?
Well, whatever's "1/2 the way" from .999 (repeating) to 1... and there you are!
(So I.E.-> Whatever the "distance/amount" is between the last repeating .9 decimal is the asymptotic amount here, and it's NEVER going to get to 1, period)
I learned 1 thing in discrete mathematics though - a LOT of what we use? Isn't EXACTLY "exact". It's a lot of "close approximations" only...
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
I'm going out later today to purchase something that cost $10 and I'll give them $9.99c and explain it's the same thing. I plan on saving a fortune using this new math.
Actually the masking of the div by zero is the whole point.
0.999*10 = 9.990 - 0.999 = 8.991
Pi for instance. As far as I can tell, it's infinite. Or 1/3 translating to 0.33333 et al. Why do these numbers go infinite? Why isn't there a definite end? And, what are the real world implications of an infinite number?
Here's to hot beer, cold women, and Glaswegian kisses for all.
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.
Finally, if you say 0.99999999..... is less than 1 : what is the difference between both?
A whole lot of ignorance.
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.
Legitimate mathematical philosophies of finitism or ultrafinitism invalidate any 0.999...=1 proofs before they can even begin.
Epsilon proofs don't work so well on forums. I prefer the geometric series approach: .99999 = 9*(.1 +.01 +.001+.0001.....)=9*(1/(1-.1) - 1) = 9/.0 -9 = 10-9 =1. I like this because practically everybody has seen it before.
a + b = b
this is true for all b if a is zero. But b doesn't need to be zero. So this equation of yours doesn't necessarily include a division by zero.
For example, let
a = 0
b = 5
then we have
(0 + 5)(0 - 5) = (5)(0 - 5)
(0-5) is not zero, and you can easily divide both sides by that to get (0 + 5) = (5)
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. -
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
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!"
Here you mix finite and infinite decimal. 0.999999...8 is NOT equal to 0.99999... because you can subtract them and get 0.00000...11111(1)..... Putting ...and something in the end is easy, but putting ... at the end is different. ;)
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.
It is relevant to note that numbers do not have unique representation even without resorting to the complexity of infinite decimal places. Consider base -2 (a perfectly good base where every integer has two possible representations). E.g., 3 (dec) = -101 (neg 2) = 011 (neg 2). Numbers exist independently of our parochial names for them, much like religion and politics are the arts of making us believe that labels, not ideas, are substance. The universe is independent of our labels and is often obscured by the labels.
Why? Because when .999... is subtracted from 10*0.999..., there is always a 0.000...09 left over.
Now if we are in the real world, 0.999... does equal 1. But we are talking about mathematical abstractions here. And in the world of mathematical abstractions, it does not. I am not aware of any proof or usage that requires this particular mathematical abstaction, but I am a physiscist and not a mathematician. In the assumed absence of a requrement for the abstraction, it is, of course entirely irrelevant and 0.999... neither equals or differs from 1. Just a different way of saying the same thing.
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.
is it just me, or is this guy smoking something.
10a - a does not reduce to 9
10*0.999... - 0.999 = 9.000999...
In case of emergency it's easier to dial 9-1-1 than 9-0.99999....-0.99999....
It's all a matter of DEFINITION.
What do you mean when you write three dots ?
AFAIK : 0,999... means : lim n-> infini of sum ( 9x10^-n)
which prooves by other means beeing equal to 1.
(then again, see what "limit" means, and what "equals" means for limit.
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
They're assuming that if a = 0.99999... then 10a = 9.9999999... and not 10a = 9.9999999...89 . You have to prove that (yes, it's obvious, but still), and that's harder (I think you need Cohen's continuum proof for that, don't you?)
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
0.0000....1 = 0 It follows from all the same reasoning for .999....=1
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.
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.
0.999 is not 1 0.9999 is between 0.999 and 1 and on and on. If you have a mass that becomes super critical and explodes at 1then 0.9999999 can't be one because of one simple fact. Electrons can't go to the next energy level unless they have the right amount of energy to reach that level. Adding more and more just means they get more and more pissed off but never angry enough to jump to the next energy level or orbit. If you take 1/3 and 2/3 and .3333 and .6666 and add them together you do get .9999... which perhaps explains entropy unless you calculate to infinity. Now if you take 1 and cut it into 3rds 1/3 1/3 1/3 and add the fractions you get 3/3 or 1 again. But if you add .3333 .3333 .3333 you get .9999 which is confusing. Maybe it means that it is a bad idea to use decimal numbers :)
I hate math. Maybe the secret to the universe is in here somewhere. Maybe there was a great nothing of 99.9999999999999999998% probability of becoming our universe, and then do to some quantum tunneling event nothingness became somethingness as the 8 changed to a 9 and the universe said screw it and rounded up to 1 and poof came into existence. Maybe this problem is in fact a side effect of the universe changing into something wholly different once we understand it to keep us always confused. (see Hitch Hikers Guide).
If anything I say is wrong it is because I have learned too much and the ideas are leaking into each other. As I gain knowledge I become dumber and dumber until one day I will know everything about nothing.
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.
This proves my theory that many numbers oscillate. The question appears to be, which ones don't?
Number that use the ... are approximations by definition. One would only look at this and think "that can't be right' if one forgets that an infinite sequence is not finite. To put that another way, to believe that .999... should be less than 1 is to believe in zeno's paradox.
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
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'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.
I think the best way to prove this to a skeptical person is to ask what 1 minus 0.9999... is. When they say "an infinite number of zeros with a 1 at the end" they should realize their error.
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
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.
All of the confusion is over people talking about two different things.
0.99999999... = 1 if you define 0.999999999... to be an infinite sequence (which converges to 1).
However, another number with the representation 0.999999999... is the number defined by the following program:
a = 0
while(true) a = 1-(1-a)/2;
They are clearly different, but in the limit they approach the same thing.
This is retarded. 10 x .999 = 9.99... it does not equal 9.999... that example proof in the slashdot post is total bull$h1t!
Isn't there room for the following?
(1-0.999999....)/2 + 0.999999...
Not sure what that is, but I can represent it in such a way that it appears to make sense.
-
The whole problem here is glossing over the ideas of infinite series and representations. Since you can't physically write digits forever, even using "..." means there is an idea sitting underneath that must be thought through and proven before you start multiplying an subtracting such objects. It's all in a freshman/sophomore level textbook, but most students don't actually "get" it- they just learn to push the symbols, which is not math. The only thing I ever asked students to carry out of my classrooms was this: Everything you write down in mathematics is the expression of some idea, and if you don't know what that idea is, you don't actually know what you're doing and you might as well write anything down that you want- you're storytelling, or maybe drawing at that point.
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
Wow... Not from the US?? Tell me more about your strange and wonderful land... Is everyone as big a jackass as you?
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.
anytime im presented with the Monty Hall problem, I only switch 99.99999...% of the time
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.
The "digit manipulation proof" assumes convergence of the sequence {0.9, 0.99, 0.999, 0.9999, ....}. The correct way to prove this is that the difference between 1 and the n'th term of the sequence is equal to 10^-n, and then for any epsilon greater than zero, this difference is smaller than epsilon for all n greater than some value N (which depends on epsilon). Thus, the series converges to the limit 1.
Note that for some decimal representations, such as that for pi or sqrt (2), the sequence does not have a limit in the rational numbers, and so we would instead have to show that the sequence is a Cauchy sequence of rationals, which has a limit in the topological completion of the rationals, known as the reals.
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
You can no more divide by 0.00...1 than you could divide by infinity. Even if you could (which you can't), that would just mean that you could divide by zero to get infinity not that it isn't equal to zero.
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.
I don't know why "relatively sharp teachers" can't see this elementary mistake. And it looks like you are their student, as you did not learn those elementary things.
I remember even in high school it would be a major mistake if you went from sqrt(a)=sqrt(b) to a=b instead of |a|=|b| or a=+/-b.
/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
What do you mean it's not a "real" number. You are going to have to use precise words here. It's just as real as .999... As to your statement about putting something after forever meaning we don't have forever, What would 10^(-infinity) be? By your logic, does that mean that infinity isn't really infinity?
I may have missed something, but how did they go from:
10a = 9.999...
to
10a - a = 9.999... - a
to
10a = 9
to
9a = 9
If there is a ? step, the least they could have done is add profit to the end.
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....
...A rational number is a number you could get by expressing as a ratio (real number divided by real number)...
I think you meant a rational number is a ratio of integers. e/pi is not rational. You are correct that any infinitely repeating decimal is a rational number, but this is always a ratio of simple integers by definition. Otherwise your point is correct.
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.
Well, I was only giving you an answer you could have "shot down" your highschool math teacher's question with, and it's correct (in theory at least)... because when he asked you to tell him a number in between .999 repeating and 1, you theoretically could have said .9991, .9992, .9993, .9994, .9995, .9996, .9997, & .9998 because they DO exist as fractional amounts on the WAY TO 1... the asymptotic material was just an example thereof (.9995 really, since it uses 1/2 way marks towards your destination everytime).
As far as what my discrete math course covered (hardest math I ever took in a CSC curriculum OR MIS curriculum in collegiate or other academia mind you, @ least I felt that way about it)? Ok, we did these areas in it:
Logic (this actually GETS tough & long, especially if you do it "the long way" vs. the possible shortcuts on truth tables, but, having seen it in philosophy before (T/F vs. Discrete & Digital Electronics 0/1 though)
Set Theory (some of what you noted I had seen in discrete, the n=k SET stuff/Epsilon symbol)
Combinatorics
Probability (this is where we got into the stuff you saw in the film "21")
SOME "number theory" (limited, very little - but INTERESTING as hell from what I recall (very little now))
SOME path type stuff (which I also really got into in other degrees as "the shortest path" algorithm for logistics & networks etc.)
Graph Theory
But, I do NOT recall what you're stating, at least not "in full" (again, some parts of it I recall... it looks like material from SET THEORY areas we studied).
DISCLAIMER: Not a "math major" here - CSC instead!
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
Heroes die once, cowards live longer.
Well there are still interesting unsolved questions related to the infinite, like the Thomson's lamp paradox. The .999... issue is more due to the limitations of notation than any deep problem.
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.
Socialism: a lie told by totalitarians and believed by fools.
physics teacher doesn't know math. (Not an uncommon ailment among physicists)
Yes, Dirac, Newton, Einstein, Rayleigh, Schroedinger, etc were all terrible at math.
I've met more than my share of mathemeticians who can't describe real systems.
The flaw in the summary's "proof" is glaringly obvious: You're conflating 10 infinite objects into 1 infinite object, remove 1 infinite object, and then -surprise!- only have a finite object left.
In reality, if you really had a collection of infinite objects, you wouldn't have 10 * 0.99999... = 9.99999..., you'd have 10 * 0.99999... = [0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999...]; and if you gave away one of your infinite objects, you'd have [0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999..., 0.99999...] - nine infinite objects.
All of these 0.99999... = 1 proofs are bullshit tricks based on imprecisions in our numeral system, either pretending that decimals could 100% accurately represent the concept of infinity, or pretending that decimals could 100% accurately represent the concept of fractions.
It's bullshit, nothing more.
1/3 = 0.33333...
3 * 1/3 = 1
3 * 0.33333... = 0.99999...
does NOT mean that 1 = 0.99999..., it just means that the decimal system has no accurate way of representing 0.0...(1/3); hence why we invented goddamn fractions - to represent that concept, because decimals couldn't properly describe them.
I really wish mathematicians and goddamn math teachers would stop wasting their time and confusing the general populace and just stfu and do real science and teaching.
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.
If this short article is correctly explaining the situation, then the math solution is wrong in the initial equation .9999... .9999.. is equal to 1, there is what is called an "Approximately Equal Sign" which can be understood on the following link, http://www.solving-math-problems.com/math-symbols-approximately-equal.html .9999... is not equal to 1, no matter how these researchers try to prove it.
Equation: 10a = 9.999
True Solution: Divide both sides by 10 to balance the equation and leave the variable alone on one side, a =
To solve, one would not subtract both sides by 'a', both sides have to be divided by 10 to leave the variable by itself on one side of the equal sign. It is simple algebra. If you want to think
These so-called researchers seem to me as if they should be "enthusiastic" about something else other than mathematics to make this idiotic argument, then write a 28 page research paper and on top of all that PUBLISH the paper! Most people commenting about this article are saying the same thing,
High school students are absolutely right to be confused when they are introduced to the 0.999...=1 "notation" or "fact" and the various "proofs" of it. For that time they have developed a pretty good sense of what mathematics is all about and of how mathematical reasoning should go according to some common sense rules (logic) by reduction to "natural" facts (axioms). Note, the axiomatic method has not been explicitly explained yet, it would come much later if ever in their learning life. The 0.999... notation is not a trivial concept it embodies the notion of infinity and of limit. It also has a complicated relation to the foundation of logic. The students are never told about these concepts, neither they have any "natural" sense of them. That is the source of the confusion, the students sense that they are cheated and they are right. They are forced to accept something baseless (from their point of current knowledge up to this point) as a mathematical fact by the same authority who until now has presented mathematics as a fair game.
(disclaimer: I am a mathematician, well aware of these educational difficulties and have not found a perfect solution to them myself.)
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.
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
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.........?
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".
Le français vous intéresse?
10*0.999 = 9.99, not 9.999.
a=0.999
wupsie!
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
a = 0,999.... | *10
10a = 9,999...0 | -0,999...
10a - 0,999... = 9,999....0 - 0,999...
(10-0,999...) * (a-0,999...) = 8,... | a = 0,999...
9,111... * 0 = 8,...
0 = 8,...
There you have it, 0 ::= 8,...
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).
I like to explain it this way:
The real number system has some weaknesses. For example, it can't express the result of division by zero, it can't express the square root of a negative number, and it can't express the smallest positive real number.
A lesser-known weakness is this: The real number system contains some numbers that do not have a unique digit sequence to represent them (for example the number that can be represented as either 1.000... or 0.999...). If you were designing the perfect number system, you would probably want each number to have a unique digit sequence to represent it. No such luck with the real numbers.
If you perceive a difference between 1 and 0.999.., then good for you. Your ability to do so will come in handy when trying to understand differential variables in calculus and how they differ from the number 0. But just note that the real number system does not have the ability to make a distinction between 1 and 0.999... like you can. (And in calculus, the real number system does not have the ability to make a distinction between dx and 0 like you can.) There's nothing "wrong" with that lack of ability, just as there's nothing "wrong" with the fact that the real number system cannot express the square root of a negative number. We're free to imagine things that the real number system cannot express, such as imaginary or non-zero-infinitesimal numbers. The key is to know the limitations of the real number system, and to understand how those limitations can create weirdness and confusion if you're not careful (like those famous "proofs" that 1=2 that arise from algebraic division by 0).
0.999 multiplied by 10 is not 9.999, it's 9.99
Therefore,
10a -a = 9.99 - 0.999
9a = 8.991
a = 0.999
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
you can't multiple 0.33333.... by 3 and get 0.999999.....
Normally, you would do this: 0.3(3) * 3 = (0.333333 + O(1)) * 3 = 0.999999 + O(1)
You always have this O(1) - to account for infinity...
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.
" 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
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.
(First, note there is an ellipsis, suggesting there are a lot more 9s after the decimal point ...)
When he multiplies by 10, he's trying to add 1 more significant digit, so everything zeros out, but he is really only shifting the significant digits over, meaning for any N decimal places in (a = 0.999...), you've got one less digit to the right of the decimal point to work with when using 10a. 10a - a = 8.999... not 9.
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.
Also a terminology issue, the number of ones in 0.111111... is indeed "Countable".
Is 10^(-infinity) meaningless? Seems like it could be 0.000...1. And if it is meaningless, is it because infinity is meaningless? And if infinity is meaningless is 0.999...meaningless?
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
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".
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
In an equation "=" means both ends of the "=" sign represent (or express, or resolve to) the same quantity. So I visualize "=" as the pivot of a scale, that's why we call it equation, I guess. So, what is people's surprise if 3 apples weigh as much as 6 eggs ?
If you please, you can read "=" in the statement 0.9... = 1 to mean the limit of the series 0.9.. which converges to 1. Again, big surprise....
The only fact I find hard to understand is why people argue so much around this issue.
Have we not all studied Math in school? Have we not all learned that "an infinite number of digit" has non practical meaning whatsoever and has not to be confused with anything you will ever be able to observe or experience in the "real" daily life?
I mean, how 0.9... being equal to 1 or not can make a difference in your life? How can you observe it? How knowing this disrupts your model of the universe? I'd be curious to know.
Please enlighten me
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.
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.
The enemies of Democracy are
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.
"The state is that great fiction by which everyone tries to live at the expense of everyone else." - Bastiat
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.
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.
Post may contain irony: discontinue use if experiencing mood swings, nausea or elevated blood pressure.
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.
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.
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).
"The state is that great fiction by which everyone tries to live at the expense of everyone else." - Bastiat
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."
Socialism: a lie told by totalitarians and believed by fools.
The so call long tail have gone where? It just disappear? or it ever exists?
Is this a hoax or something?
It says a = .999 and 10a = 9.999 but 10a should be 9.990 !
Regards,
Vishal
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.
let a = b
multiply both sides by a
a * a = a * b
subtract b^2
a^2 - b^2 = ab - b^2
factor out a - b
(a + b)(a - b) = b(a - b)
cancel factor
(a + b)(a - b)/(a - b) = b(a - b)/(a - b)
a + b = b
a + a = a
2a = a
2 = 1
same flaw - easier to follow (and remember)
Isn't it just a bad multiplication with infinity involved: a infinity of nines multiplied with 10 is still a infinity of nines. So the infinity of decimals are the same.
just think about it a bit, ie when using finite numbers for a:
a = 0.999
10a - a = 10*0.999 - 0.999
10a - a = 9.99 - 0.999
9a = 8.991
a = 0.999
Let be:
STUDY = NO FAIL
NO STUDY = FAIL
So
STUDY + NO STUDY = FAIL + NO FAIL
STUDY (NO +1) = FAIL (NO + 1)
Then divide by (NO + 1)
STUDY = FAIL!
(seen on a T-Shirt)
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.
Socialism: a lie told by totalitarians and believed by fools.
0.999 does not equal 1. First, multiply both sides by 10 does not equate to 10a = 9.999 (it equates to 10a = 9.99). Next subtracting "a" does not equate to 10a - a = 9.999 - 0.999 (remember that it's 9.99). So when you do subtract "a", it equates to 9a = 8.991 and divide by 9 equals a = 0.999. As it should be.
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
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.
Why does 9.9999999..... - 0.999999.... = 9
The first term came from 0.9999999... x 10, so in the infinity decimal place, it has a zero, not a 9, so there's a 9 left.
They just left-shifted the number and somehow pulled an extra 9 off the right side of infinity.
( 0xFF 1 ) - 0xFF = 0x100, when the shift fills with 1 and 0xFF1 is 0x1FF
( 0xFF 1 ) - 0xFF = 0xFF, when the shift fills with 0 and 0xFF1 is 0x1FE
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.
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
.3333...=.3333... .3333...=1
a=.3333...
10*a=10*.3333...
10*a=3.3333...
10*a - a= 3.3333... - a
3a=3
a=1
am I missing something?
is it just me?! the obvious flaw is that 10a is NOT 9.999... it's only 9.99! that's were the OP fails.
pc.
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?
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
Instead of dealing with fractions of 3 and 9, here is a way of looking at it which can't be 'disproven' because the indefinite representations of the fractions supposedly aren't exactly the same as the fractions themselves.
Of course, it does carry its own stigma with it in the fact that it uses perverse division. But I am welcome for anyone to point out an actual problem with the process which isn't 'that isn't the way you are supposed to do it!'.
This does also rely on the 'assumption' that a number divided by 1 will continue to equal itself.
In this situation, we are dividing 1 by 1.
Instead of taking the 1 out of the 1 as it stands, we carry the 1 to the next column. Here I will be using parentheses to indicate a number with its carried value, not to indicate a repeating number.so we're now doing 1 into 0.(10).
Take 9 out of that and carry the 1 to the next column. Our result is currently 0.9 with work left being taking 1 into 0.0(10).
Take 9 out of that and carry the 1 to the next column. Our result is currently 0.99 with work left being taking 1 into 0.00(10).
Ad nauseum, you can see that the result of the division 1/1 is 0.999 repeating.
So, I would like for someone to point out a problem with this proof? Or do the naysayers claim that regular 0.999 repeating is a different 0.999 repeating than the result we get here?
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.
Object Oriented languages are just languages. Kind of like English, but simple enough that machines can understand them. Nobody claimed that OO was the formal basis of mathematics, not that machines actually represent numbers as OO classes.
The formal basis for reasoning about the reals would be to take a formal definition of them such as "a Dedekind-complete ordered field" an go from there. If you are interested in the actual definition of the reals a good place to start would be to look at axiomatisations of the reals
That is stupid because whole numbers are discrete and real numbers are continuous.
Is this just a neat way to introduce elipses to 3rd-graders, or do any other results in mathematics depend on this?
No, because an elipsis is the wrong way to denote an infinitely repeating series of numbers, but since slashdot has trouble with special characters it convenient to use it here.
It is wrong because you can have more than one repeating digit (e.g. 14/11) and you need a way to mark each digit that repeats. I seem to recall the notation as being a dot over each digit that repeats, but I could be mistaken (been a long time since I did it at school), someone else here in this discussion said it was a line over each digit that repeats.