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*

Simplify

*9a = 9*

Divide both sides by 9

*a = 1*

*Therefore .999999r = 1*

## 1 comment:

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...

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