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

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

[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

[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:I went one further

[MyLongNickName]

And seriously... is this really front page material? The simplest proof is to say "express 1/9" as a decimal. Now multiply both sides by 9. I remember this in elementary school algebra.

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.

Or

[Anonymous Coward]

1/3 = 0.3333...
2/3 = 0.6666...

0.3333.... + 0.6666.... = 0.9999....

1/3 + 2/3 = 1 = 0.9999.....

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:Finally

[SoVeryTired]

If pressed, many logicians will admit that the modern foundation of mathematics (ZFC) is probably inconsistent.
See this article:
http://www.math.princeton.edu/~nelson/papers/warn.pdf

The author discusses an informal survey he took among loogicians on page three.

If someone ever discovers a paradox, we can simply scale back to some other system and keep most of what we know, but still...

Re:I went one further

[cgenman]

Conceptually, 0.999... keeps getting closer and closer to 1, as you add more decimal places. It approaches 1. This limit is how all calculus works. Any series that approaches another number as you flesh out the series further and further, will be that number once you have taken the series to infinity.

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:Finally

[Jason Levine]

Suppose you have 3 numbers, a, b and c such that c = b - a.

Multiply each side by (b - a) to get:

c(b - a) = (b - a)(b - a) => Or....
cb - ca = b^2 - 2ba + a^2 => Now add (ab - a^2 - cb) to both sides
ab - ca - a^2 = b^2 - cb - ba => Or....
a(b - c - a) = b(b - c - a) => Divide both sides by (b - c - a) and.....
a = b

There you go! Proof that any two numbers (such as 0 and 1) are equal.

(Yes, I know there's a flaw in there. Let's see who'll spot it first.)

Re:Finally

[drakaan]

Okay, but this isn't a problem with the foundation of math being inconsistent, this is a problem with people not knowing how to write the number normally known as "1" in a different way. Most people would grasp "3/3" as being the same as 1, but this *looks* different because they're unused to seeing it.

The fact that the fractions 1/3 (known in decimal notation as .3...) and 2/3 (known in decimal as .6...) have a sum that can be written funny doesn't mean that they don't still add up to 1.

A mathematical amusement causes people confusion and consternation. It's like asking someone why they appear reversed left-to-right in a mirror, but not top-to-bottom, and saying there's an inconsistency in the foundation of physics.

The problem is that partial understanding of a subject and an associated problem in that subject makes things *appear* inconsistent when they are not.

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.

I've had this argument more times than I'd like

[Benfea]

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.

I've tried what you suggest, and it DOESN'T WORK

[Benfea]

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.