This question had bothered me for fifteen years already

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I have wondered about this seeming inconsistency since I first start learning fractions in elementary school. Most people I ask just look at me like I asked them which one of my testicles is heavier. What is the difference between 1 and .9 repeating?

1/9=.1 repeating
2/9=.2 repeating
3/9=.3 repeating
4/9=.4 repeating
5/9=.5 repeating
6/9=.6 repeating
7/9=.7 repeating
8/9=.8 repeating
9/9=.9 repeating or 1 ???

Ok, another approach.
.1 repeating + .1 repeating (1/9+1/9) =.2 repeating (2/9)
.4 repeating + .1 repeating (4/9+1/9) =.5 repeating (5/9)
.7 repeating + .1 repeating (7/9+1/9) =.8 repeating (8/9)
.8 repeating + .1 repeating (8/9+1/9) =.9 repeating

How is this?

.999... = 1
There is no difference.

We could check this using a geometric sum
.999... = 9/10 + 9/100 + 9/1000 + ...
= (sum from i = 0 to infinity of 9*1/10^i) - 9 = 9(sum from i = 0 to infinity 1/10^i) - 9 = 9(1/(1-1/10)) - 9 = = 9(10/9) - 9 = 10 - 9 = 1
QED

Yes, and another good proof.

a=.999...
10a=9.999...
10a-a 9.999...-.999...
9a=9
a=1

I've read them all. Both of my other demonstrations were informal proofs that you are right. Everything says I'm wrong. I just refuse to accept it. How can there be more than one 1?

It is not that there is "more than one 1"; .999... is just another way of writing 1. The difference is entirely notational.

I may just make rekoil's day.

There is no reason not to define a number system where 1.0 and 0.9999... are distinct. To be even sillier, you could choose to make 1.0 and 1.00 be different and there's no obligation for either to be the same as a plain 1 without a decimal point.

Unfortunately, such systems aren't very useful.

The mapping of the real numbers to the decimal (or binary, or sexagesimal) representations happens to be where the identity of 1.0 and 0.99999.... comes in. They are two ways of writing a decimal expansion for the same real number.

In fact, when using decimal expansions, every real number that is representable by a FINITE decimal expansion (i.e. is of the form n/10^m where n and m are integers), has TWO decimal representations. The obvious one, which is actually followed by an infinite series of zero digits, and the less obvious one, which is the finite (n-1)/10^m, followed by an infinite series of 9 digits.

there is a number system called the hyperreal numbers in which 0.99999 recurring doesn't equal 1, (or to be more accurate there are infinitely many hyperreal 0.99999 recurrings, most of which don't equal 1 )

it is extremely useful, mostly because it makes maths easier to do