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

1 comment:

alfredo said...

TJ, I found your article very interesting. But as far as i know the "closest to reality" explanation (if there is any reality in math) is that
there is NOT a human readable number that results from dividing a number by 9. So, the .XXXXX that we get is not the *real* answer but one that we can use in real life.
So what you actually found is NOT that .999r = 1 but a paradox made by this way of representing unrepresentable things.
If you use a professional software you will notice that it never converts fractions like 9/7 unless you actually force it to.
Any way, you are right that 9/9 is the same as 9.9999 plus an infinitesimal which is more like an philosophical entity rather than a number so kinda...