This morning I was reading slashdot and came across a story about some math enthusiasts that proved 1=.999.. The gist about how they did this is as follows.

a=.999..

10a=9.999..

10a-a=9.999.. – .999..

9a=9

a=1

Seems kind of strange. So inspired by this I set out to duplicate it with another similar situation.

a=.333..

10a=3.333..

10a-a=3.333.. – .333..

9a=3

a=1/3

In this case however we end up with the exact fractional representation of .333.. which got me thinking. Maybe the problem in the original is that .999.. isn’t representable as a fraction. For instance, using similar logic as the original proof:

.333.. = 1/3

3*.333.. = 3 * 1/3

.999.. = 1

However, in this example, it’s obvious the issue is an error. .333.. * 3 doesn’t equal .999.., it equals 1. So looking at the original proof, if instead of a, we use a constant for .999.., say n, and we treat it like we treat any constant, then the proof looks like this.

a=n

10a=10n

10a-a=10n – a

9a=10n – n

9a=9n

a=n

Nothing out of the ordinary really happens. The only reason anything strange happened last time is they authors are performing arithmetic on numbers that don’t behave as they expect. The whole thing reminds me of a proof about 1 = 2 that gets to the result by dividing by zero.