Saturday, March 12, 2011

When .999 is actually 1

You remember at school (or maybe you're just learning now) when doing fractions, that dividing something by 9 would give you an "infinite" decimal that repeated? For example 4/9 = .444444 on to infinity.

This means that if you ever see a number that is that way you know it's something divided by 9, so .77777...  is 7/9.

But wait, what about .999999... ? That would be 9/9 which is 1, that can't be right!

In fact it is, and without any tricks, here's the algebra to show it.

I will use .999999r to indicate .9 repeater, normally it would be a .9 with a . above the 9, but, finding a font that everyone has that has that proved too tedious :-) I have also made all the steps explicit instead of doing multiple operations at once, just so algebra beginners can see there is no subterfuge going on.

Let a = .999999r

Multiply both sides by 10 gives
10a = 9.999999r

Subtract 9 from both sides gives
10a - 9 = .999999r

Replace .999999r with a from above
10a - 9 = a

Subtract a from both sides
10a - 9 - a = 0

Add 9 to both sides
10a - a = 9

9a = 9

Divide both sides by 9
a = 1

Therefore .999999r = 1