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*