proof for the general rule of conversion from base 10 to other bases I just begin reading the book "what is mathematics" by Richard Courant. He states the general rule for passing from the base ten to any other base B is to perform successive divisions of the number z by B; the remainders will be the digits of the number in the system with base B. This is very intuitive and I understand the rule. But How do we prove it? this has raised a very important question: how do we prove intuitive concepts? Any suggestions?

This rule comes straight away from the definition of base.

Suppose you are given a positive integer $x$. It doesn't matter how it is written, as long as you can identify the number (in particular, the original number system doesn't matter). Now, you want to write it as $$ x=\sum_{i=0}^{n} b_i B^i $$ where $\\{b_i\\}$ are the digits of $x$ in the base $B>1$. Observe that if you divide $x=x^{(0)}$ by $B$, by Euclidean division you get $$ x^{(0)} = r_0 + Bx^{(1)} $$ for some (positive) integers $x^{(1)}
You can then easily adapt this to cover negative integers and real numbers, too. Note that for a generic real number a finite number of divisions won't be enough, though.