0.99... = 1

[segadragon]

Some may have heard of this, but for others... it's time for a major lesson.

0.99 repeating. Meaning 0.9999999999999999999999999 (where 9 goes on forever).

This is the same number as 1.
There are three ways to prove this. I can only remember two. Anyone who knows the other, be sure to post it.

Method A)
If you don't know what 1/9 is in decimal form, put it in your calculator.
It's 0.11... (the ... will be used to represent "repeating" throughout this topic).
This would mean 2/9 is 0.2... (It would be 0.1... times 2)
3/9, or the infamous 1/3, we should all know is 0.3...
4/9 = 0.4...
And I'm sure you're catching my drift, right? Make sure to put these in your calculator, so you can see for yourself. Let's keep going for math's sake.
5/9 = 0.5...
6/9 = 0.6...
7/9 = 0.7...
This is so easy, right? What's my point?
8/9 = 0.8...
So what is 9/9? Duh! 0.9...!
But I thought nine out of nine was one?

Method B)
Do you know Algebra? Good.
We'll start with variable X.
X is equal to 0.9...
X = 0.9...
(so what's 10X?)
10X = 9.9...
(That was easy, right? Just move the decimal place. Simple first grade stuff, here; I wonder what happens when I subtract X from each side)
10X-X = 9.9... - 0.9..
9X = 9
(Hmm... Let's divide by nine for no reason at all)
X = 1

Thank you for reading.

[realillusion]

Neither of those constitutes a proof, though they might server as useful illustrations...

There are plenty of proofs that are accessible though.
"Method A" at least bordered on a proof. If you had instead endeavored to show that
.A_bar = ∑(A*1/10^n; n=1; infinity)
Easy enough with Lemma: ∑(r^n; n=a with a>0; infinity) with |r|<1 = 1/(1-r)

A simpler proof is done by contradiction: assume that .9_bar and 1 are unique values. By basic axioms for a complete ordered field, there has to be a value x such that .9_bar < x < 1, but we know of none.

You can also easily apply limit theories, and/or L'Hopital. That might not be as accessible, but I throw the option out there for those who are familiar with it.

[segadragon]

Hey. I finally got that .A_bar part. It just says 0.9_bar = 0.9+0.09+0.009+0.0009...
Well, golly. I sure am learning.