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Proving 0.999... Is Equal To 1
eldavojohn writes "Some of the juiciest parts of mathematics are the really simple statements that cause one to immediately pause and exclaim 'that can't be right!' But a recent 28 page paper in The Montana Mathematics Enthusiast (PDF) spends a great deal of time fielding questions by researchers who have explored this in depth and this seemingly impossibility is further explored in a brief history by Dev Gualtieri who presents the digit manipulation proof: Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1."
(0.999...)st Post!
I was able to prove that with even one less "9" after the decimal point, it STILL equaled 1. I plan on doing this for a few more iteration until I can prove that . = 1
See my journal for slashdot ID's by year. Mine created in 2005. http://slashdot.org/journal/289875/slashdot-ids-by-year
Someone disproved math. Kids around the world celebrating. Accountants are lighting themselves on fire. Corporate greed accellerates. 'Office Space' now seen as a prophecy.
0.999... = 1 is second place to the Monty Hall Problem on the list of things that people have difficulty understanding and accepting the proof of. It is second place because the only department where I do not see graduate students giving me a confused look is the math department; with the Monty Hall problem, I will sometimes get a confused look even from people in the math department.
The other reason I put it in second place is that most people have difficult understanding the problem at all, whereas very few people have trouble understand what the Monty Hall problem is asking.
Palm trees and 8
Now I can replace my SLA with 100% uptime.
1/3 = 0.3333...
2/3 = 0.6666...
0.3333.... + 0.6666.... = 0.9999....
1/3 + 2/3 = 1 = 0.9999.....
They damn well better, how else will I measure out all this dental floss.
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.
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.
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.
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's a real crowd-pleaser at parties. Personally I like the "writing an executable java program without a main method" trick to impress the ladies myself--that is, if I ever get to meet an actual lady who would even get that trick.
SJW: Someone who has run out of real oppression, and has to fake it.
But you cannot reach infinity so this is a moot point.
I think you just dismissed most of mathematics.
Actually, if you define 0.999... as having an infinite number of decimal points, then it is true. And that's how that ellipsis is defined! It means exactly infinite repeating decimals.
You've demonstrated the first hurdle that this problem raises in people's brains: they start thinking about adding "one more" decimal point to the expression, meaning they're thinking of a large but finite number of decimal points. And the second hurdle: people find it hard to believe that you can do mathematics with "infinity" as a meaningful quantity.
No kidding!!! What do you say at this point?
I wasn't trolling :(
0.999.... might be equal to 1, but 0.999 is not equal to 1:
x=0.999
10x = 9.99
10x-x = 9.99 - 0.999
9x = 8.991
x = 0.999
This is true whether there's three 9s or a hundred 9s, so I can see the confusion.
my karma will be here long after I'm gone
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.
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The series is infinite, you don't lose one.
Just because you can not show the number as a whole does not mean
you can not perform operations using it.
i.e. Think of pi.
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.
Oh... They aren't empty. The aliens live in them now. They think the high radiation is good for their complexion.
No, Ziaxia, I wasn't telling them anything on slashdot, GET OUT OF MY HEAD! GET OUT OF MY HEAD! AHHHHHH!!!! Don't make me explode! ^h^h^h^h^h^hcarrier lost
Self proclaimed typo king, and inventor of the bear destroying coffee table (patent not pending).
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.
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!
I'm not even close to a mathematician, so forgive this possibly very stupid question:
.999... by anything at all? If the sequence is infinity, then any application of task or step can never be completed as it would take infinite time to perform the calculation.
How can you multiply
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."
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders a half a beer. The third orders a quarter of a beer. The bartender says, "You're all idiots," and pours two beers.
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.
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?
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.
Infinity gets weird as hell. I don't remember the exact proof, but in a calculus class in college, the professor posed this question for "fun": if you have a ball that recovers 9/10 (or whatever the figure was, it's been a while) of its height with every bounce, how long will it take it to come to a complete stop? One of the students worked it out before the next class session, and we were all amazed to learn that even though the ball will make an infinite number of bounces, it will do so within 30 seconds. So an infinite number of actions can, theoretically, be performed in a finite amount of time. Pretty crazy, but the math was sound.
"16MB (fuck off, MiB fascists)" - The Mighty Buzzard
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.
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
This is absolutely preposterous. Of course you can add 1/9 to 1/9. The answer is 2/9. You are failing to separate what can be added as floats in base 10 with what can be added at all. For example, there is no reason that we cannot represent 1/9 in base 9 as 0.1, then add 0.1 to 0.1 to get 0.2 in base 9. This can all be done without approximation using floating point arithmetic, just not in base 10.
The only reason infinity might come into play is because 1/9 has no finite representation in base 10. But this is not a problem. Consider the number 1/10. Would you say that you can multiply 1/10 by 10? Assuming that you would agree to this, consider what would happen if you performed this operation in binary? In base 2, 1/10 has no finite representation, so by your logic, we cannot multiply it by 10. In fact, given any finite decimal there is another base in which it has only an infinite representation, and hence by your logic we cannot multiply numbers by decimals ever.
You have imposed an artificial limitation on the basic concepts of arithmetic by limiting yourself to floats in base 10.
I also have a problem with you stating that you can't use the value of pi in math when you absolutely can. The inability to write down infinitely many digits (by which I mean write down more digits than any finite number of digits) does not preclude the fact that we can do math with pi. Math is in no way limited to calculations which can only be done with finite precision. Take this example from calculus: find the area under the curve 4/(1 + x^2) from x = 0 to x = 1. We can show that the antiderivative of 4/(1+x^2) is 4arctan(x), and so the answer is 4arctan(1) - 4arctan(0). By the definition of the tangent function, tan(pi/4) = 1 and tan(0) = 0, so 4arctan(1) - 4arctan(0) = pi. That is, the area described above is exactly pi, or exactly the same as the area of half a unit disk (even though these regions look nothing alike). Notice that the answer is not that these are pretty close, they are in fact so close that given a set of tools with any arbitrarily small margin of error, we could not tell them apart. This is just one example of doing math with pi and not a definable approximation of pi.
Also a terminology issue, the number of ones in 0.111111... is indeed "Countable".
You fail at subtraction.
1 - .111... is not equal to .999...
Try this: .1 = .9 .11 = .89 .111 = .889 .111... = .888...
1 -
1 -
1 -
1 -