When I was very young and new to math I remember being very fascinated with repeating decimals, things like this

\[0.121212121212...\]

for the rest of this post I will use the bar notation and represent the above as this which is the same thing:

\[0.\overline{12}\]

Anyway what I found curious as a kid was that you can represent an infinitely repeating decimal as a fraction where it is just the repeating bit of decimals over the same number of digits of 9. So the fraction form of the above looks like this

\[\frac{12}{99}\]

This blew my mind as a kid, it seemed almost magical how denominators of multiples of 10 were for "regular" decimals and denominators of 9 were for repeating decimals. What is this voodoo!

Of course it didnt take me long to put two and two together and realize that the fraction \(\frac{9}{9}\) results in 1 and not \(0.\overline{9}\) as the above naive rules might expect.

Of course I went to the math teacher to try to get some help understanding this and having a math teacher try to explain to a kid in first or second grade that \(0.\overline{9}\) is actually the same as \(1\) wasn't easy for him I'm sure.

I remember his logical point was "well you cant have an infinitely small number, so if the 9's go on forever they are just approaching 1". As an adult looking back I realize he was thinking of it in calculus terms, which is all well and valid, but it didnt translate too well to a little kid. I remember coming back tot he teacher the next day, smug as hell, ready to prove him wrong and I came up with this equation:

\[0.\overline{9} = 1 - 0.\overline{0}1\]

He said "you cant do that" I replied "I just did!"

As an adult knowing limits and all that I understand what the teacher was saying, and of course he was right in a way. But then even older as I got deeper in math I learned of the concept of an infinitesimal, which is basically the very idea I was conveying as a kid just in a less refined way, without good wording. So now I look back and dont know if my teacher was outsmarted by child-me, or what.