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."

Comments

(0.999...)st Post!

[Anonymous Coward]

(0.999...)st Post!

Re:(0.999...)st Post!

[Dan East]

Geez these first posters. Like spammers, always looking for a new attack vector. I'm sure he's been sitting on this particular exploit for a long time, just waiting for his opportunity to strike. You've won today, but we're all onto your trick when you try to (0.999...)st post the next story...

Re:(0.999...)st Post!

[derrida]

2+2=5 for sufficiently big values of 2.

or for sufficiently small values of 5.

Re:(0.999...)st Post!

[Vintermann]

But, if you choose the rounding method known as "floor", then 0.999... is 0, right? So for sufficiently bad rounding methods, 1 = 0.

Re:(0.999...)st Post!

[interkin3tic]

Shoot, I just spent my last 0.999... mod points.

I went one further

[MyLongNickName]

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

Re:I went one further

[betterunixthanunix]

Small numbers usually win; express 1/3 as a decimal, and multiply by 3. The problem with that, though, is that people have trouble accepting that there was nothing wrong with what they did -- a lot of people have this implicit assumption that if a few simple steps bring them to a result that doesn't look like it makes sense, then they did something wrong. If you get them a more complicated proof (assuming they can follow it), they are more willing to accept the result.

Re:I went one further

[MozeeToby]

a lot of people have this implicit assumption that if a few simple steps bring them to a result that doesn't look like it makes sense, then they did something wrong.

It should be noted that this is not a bad thing. Indeed, it is one of the first things that a good math teacher will teach to the class - all answers should go through a 'does this make sense?' filter before you consider the problem done. It is only very rarely that it causes problems, and it is exceedingly common that it prevents them.

Re:I went one further

[radtea]

The problem with that, though, is that people have trouble accepting that there was nothing wrong with what they did -- a lot of people have this implicit assumption that if a few simple steps bring them to a result that doesn't look like it makes sense, then they did something wrong.

Nope, the problem is that the people who discuss this question are lousy teachers. They set it up deliberately to create a block in other people's minds that makes it unnecessarily difficult for them to understand what is being claimed and why it is true.

If instead they said, "It is possible to represent numbers in different ways. We all know this, and it's completely uninteresting, but I'm going to bore you with it anyway. You know you can represent 1/3 as 0.3333... right? No big deal. Now curiously that also means you can represent 1/1 = (3*1/3) as 3*0.3333... or 0.99999... It's just a different representation of exactly the same value. You can of course also represent 1 as 5*1/5 1/2+1/2 and all kinds of other awkward and unintersting ways, too."

I'm not sure why people insist on presenting this result in the most counter-intuitive way possible and then wasting vast amounts of time trying to undo the damage they've inflicted with their incompetent introduction of the problem. My guess is that they are simply not very smart, as anyone who isn't fairly dumb would see that there is an obvious pedagogical problem at play here, and correct their presentation accordingly, rather than blindly and stupidly repeating the rote "0.9999... = 1" introduction to the remarkably dull fact that you can represent the same value in different ways.

Of course, in an insanely strictly typed language with infinite precision 0.999... would not quite be the same as 1, as the former is a real and the latter is an integer, so despite having the same value their different types would mean they could not be used identically in all circumstances.

Re:I went one further

[MobileTatsu-NJG]

And seriously... is this really front page material?

You'd rather argue about smartphones?

Re:I went one further

[Evo]

Surely the problem is that you're assuming sqrt(1) = 1 when actually it is +- 1? You're throwing away the sign change in that step :)

This is second place

[betterunixthanunix]

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.

Re:This is second place

[MyLongNickName]

It is easy to explain.

1. 1/9 = 0.111111111111111111111111111111.....
2. Multiply each side by 9
3. 9/9 = 0.999999999999999999999999999999......
4. Simplify fraction
5. 1 = 0.999999999999999999999999999999......

Monty Hall trips up even serious math enthusiasts.

Re:This is second place

[ObsessiveMathsFreak]

The Monty Hall problem and its delinquent cousin the Tuesday Boy problem are genuinely difficult because the answer is highly dependent on the way that the question is posed.

0.9999...=1 is not genuinely difficult because at the end of the day it's a very informal statement about adding an infinite number of decimals, and the only real controversy about the statement exists among 4chan trolls and Wikipedia users. Most who don't understand don't care and most who do understand also don't care.

The only people with a problem are the people who don't understand but still care, but then that's the problem with most topics these days.

Re:This is second place

[kannibul]

This could be done with any fraction represented as a repeating decimal.
The trip-up is that it's repeating...since we have no concept for infinity, and, that there's no method of resolving a fraction w/ repeating decimal...it's not an accurate representation of the fraction - that's the flaw.
Therefore, Fractions are Good. Decimals are Evil!
Good thing our banks, credit card companies, and governments don't use repeating fractions.

Re:This is second place

[Colonel Korn]

Unfortunatelly, your proof is not valid. You are trying to prove something which you postulate in your first step.
How do you know 1/9 equals 0.1111111.... ?

He begged the question! For anyone confused about the term "beg the question," this is exactly what it means: assuming the proposition to be proved in the premise.

But that begs the question: is the classical meaning already dead, replaced with the much more easily understood modern usage demonstrated here?

Re:This is second place

[Missing.Matter]

I remember being told this in highschool. There was much objection, but the teacher shut us up by simply saying "give me a number in between them."

Re:This is second place

[metamechanical]

The Monty Hall problem and its delinquent cousin the Tuesday Boy problem are genuinely difficult because the answer is highly dependent on the way that the question is posed.

I would argue that the Monty Hall problem is difficult because people don't take into account the fact that the result is NOT path independent.

It would be much easier (I think) to understand intuitively if people realized that it was highly likely that they picked the wrong door to start. A more intuitive way of explaining the problem to somebody would be to increase the number of doors - to say, infinity. If there are infinity minus one closed doors with goats behind them, and a single door with a car behind it, the odds are obviously very high that you picked a goat. The probability that you picked the car is vanishingly small. Therefore, when the host opens every door except yours and one other, and they all reveal goats, the odds are very, very high that the other door hides a car, and yours hides a goat.

Now, reduce that to 3 doors. The same logic applies.

Re:This is second place

[Johnny Mnemonic]

Typical engineer. Here's the operations perspective:
a reliability of 1.0 equates to never fail.
a reliability of .999... means "sometime fail".

The sales guy will sell 1.0, and when failure happens, explain that what was really meant was .999...

Good luck with that.

Re:This is second place

[Chapter80]

I remember being told this in highschool. There was much objection, but the teacher shut us up by simply saying "give me a number in between them."

Duh. 0.9999... and a half!

Time to Update my SLA

[Anonymous Coward]

Now I can replace my SLA with 100% uptime.

Re:Finally

[Vectormatic]

just as long as no-one proves 0 = 1 we computerpeople are safe...

Cribbed, Since My Memory for Jokes Sucks

[Anne_Nonymous]

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."

Re:Cribbed, Since My Memory for Jokes Sucks

[Daniel_Staal]

Actually, a good physicist should have been able to give an answer (or something close to it) as well...

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.

Re:This is just faulty math

[Sockatume]

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.

Re:When you add/subtract/multiply/divide infinite

[BlackPignouf]

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

The decimal expansion of a rational number always either terminates after finitely many digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.

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.

Re:Finally

[MyLongNickName]

I am compelled to answer...
Divide both sides by (b - c - a) is dividing by zero.

Corrected, Since My Memory for Jokes Sucks

[Anne_Nonymous]

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."

Re:Humans are just biased towards natural numbers

[blueg3]

They're not proving "0.99999 = 1" at all. That's not true. They're proving that "0.999... = 1". One is an infinite sequence of digits, and the other isn't. The distinction is important. The proof of "0.999... = 1" has nothing to do with rounding, and to suggest so indicates a (common) gross misunderstanding of the problem.

First, you only measure things with such poor precision because you're working well above the quantum level.

Second, natural numbers are certainly important. For one, they're critical to our understanding of the rest of mathematics, which is important for fancy things like being able to take measurements and manipulate them at all. For another, we work with whole numbers of objects all the time -- two apples, ten antelope, four huts, etc. It's not "10 +/- 0.01 antelope".

Re:Cat and Mouse

[blueg3]

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.

Re:I've tried what you suggest, and it DOESN'T WOR

[Omestes]

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.