ok so i understand it's impossible to represent .01(decimal) in binary...an example i know of would be just like how 1/30 is impossible to represent in decimal

1/30 <--- always gives u .033333.....it goes on forever and i can even prove this by writing out 1 divided by 30 and i notice i keep getting the same number over and over and i am in an infinite loop. my question is how can i prove that .01 cannot be represented in binary form,without using a repeating decimal division problem? in other words i can i prove this by USING only binary digits to get my point across?

is .01(decimal) even representable in binary?

Take a look here.
Think about it - if you have floating points on your computer, then there is a binary representation of it :)

Another way to help you think about this problem. If you know calc then this will make more sense. Think about it this way.

1. you are on a number line, specifically a segment from 0 to 0.01
2. the idea is you are taking an infinite number of half steps to 0.01 (ie, first step gets you to .005, 2nd to 0.0075, 3rd to 0.00875...nth step to 0.01-(0.01/2)^n) each "step" is 1 bit
.*. and you can only get to 0.01 as n->infinity