For anyone who is familiar with math headaches, this is one of the major migraines. You can easily find all manner of argument over the internet as to the matter. I am not going to claim that in one short blog entry I can once and for all close the matter, but I can at least refute one argument.

Specifically, the following argument.

The precise mathematical proof consists of assuming first a number N which is defined by:

1) N = 0.999999999...

If we now multiply both sides of the equation by 10, we obtain:

2) 10N = 9.999999999...

Now substracting N from each side of the equation, we obtain:

3) 10N - N = 9.999999999... - N

4) = 9.999999999... - 0.999999999...

5) 9N = 9

or

6) N = 1 = 0.999999999...

It seems pretty conclusive, but there is a subtle mistake that is made at 5, specifically in the subtraction on the right hand of the equation. It should read as follows.

5) 9N ≈ 9

or

5) 9N = 8.9999999...9991

This is because trailing the end of 9.999... is an additional 0 (due to multiplying by 10), while trailing the end of 0.999... is a 9. That single digit difference is what unravels the issue. Either it must be admitted that step 5 is only an inaccurate approximation made because we are too lazy to actually travel down the path of infinity to find the "last" digit, or we have to account for the difference and display the equation appropriately.

And in case you were wondering, .8999...9991 / 9 = 0.999...

Another common argument is the table of nines.

1 / 9 = 0.111...

2 / 9 = 0.222...

3 / 9 = 0.333...

...

8 / 9 = 0.888...

9 / 9 = 1 = 0.999... = 0.111 * 9 = 1 / 9 * 9

Again, this is slightly disingenuous but in a slightly different fashion. Rather than ignoring the relationship of two separate infinites, this one ignores a fundamental concept learned in grade school. Specifically, the concept of remainders.

Trailing the edge of 0.111... isn't a pure 1, it's a 1 R(1/9). Evaluating the remainder is what gives us our next decimal place in infinity, but it will also, always have it's own remainder of 1/9. This remainder is unrepresentable in machine terms/thinking without evaluating it, hence infinity and the erroneous concept that 1 / 9 "ends" in a simple 1.

It is this remainder that, when multiplying 0.111... by 9, evaluates to 0.000...0001 and ticks the value over from 0.999... to 1.

Q.E.D. I am a nerd, thanks Dad.