Does 1=.999...?

I heard a joke about logic a while back:

One man's modus ponens is another man's modus tollens

Though, I am hesitant to tell such a joke, for fear it will injure someone through uncontrollable side-splitting laughter. I am not responsible for burst spleens! ;)

It basically means any valid chain of reasoning can be run in two directions. For example:

Here's one proof for 1 = .999...


In Axiomatic System 1

(1) Assume: 1/3 = .333...
(2) Then adding this equation to itself three times results in:

1/3 = .333...
+ 1/3 = .333...
+ 1/3 = .333...
-------------------
1 = .999...

(3) Therefore 1 = .999...

However, this result seems a bit counter-intuitive.

There is however an equally valid argument that can be constructed by running this same chain of reasoning in reverse:


In Axiomatic System 2

(1) Assume: 1/3 = .333...
(2) Then adding this equation to itself three times results in:

1/3 = .333...
+ 1/3 = .333...
+ 1/3 = .333...
-------------------
1 = .999...

(3) Therefore, 1 = .999...
(4) However, clearly this conclusion is absurd. There is one point of difference, if we define the set of real numbers by repeatedly dividing a line into ten parts. The coordinate .999... would describe the next-to-last point in the line.
(5) Therefore we can conclude that the assumption (1) was false (by reductio ad absurdum).
(6) And since 1 > .999... by one point,

1/3 > .333...

Just like that we get a completely different conclusion.


Point Schmoint

Generally both axiomatic systems coincide on calculations, literally, 99.9... % of the time. Does a difference of a point or two matter? Who would notice the error in a calculation off by such a small amount?

The practical significance will emerge in a calculation that amplifies the difference, such as:

(1 - .999...) * (the number of points in a line of length 1)

This infinite summation is an indeterminate form, roughly:

0 * oo

In the first axiomatic system we may take this product to be 0. Since if the difference is truly zero, zero times anything must be zero.

In the second axiomatic system we may take this product to be equal to 1. Since if N represents the non-finite number of points in the line, 1/N * N = 1.

Neither 0 or 1 is "the" correct answer, any more than saying Checkers is "more correct" than Othello. Different ground rules will lead to different results in different games. A chain of reasoning only establishes a connection between a set of assumptions and the conclusion. It does not establish a direction the reasoning must travel along that chain. Just as we can accept the conclusion, we may equally reject both the conclusion and at least one assumption which led to it.

We might even imagine a system in which 1=0 (mod 1, for example), and both answers are correct. :)

So overall, we may take these numbers '1/3' and '.333...' to be infinitely close together on the number line, but not exactly equal. Or we may take them to be *exactly* equal -- the exact same point. Both systems can be consistent. But whether or not they are equal really depends on what our definition of "is equal" is.