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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?

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Your second question is way too broad. Note that some intuitive concepts are easily proved (like the one which started your question), but others are extremely hard (just think of Russel's Principia Mathematica) or plain wrong (like the Mertens conjecture). – A.P. Apr 20 '13 at 11:15
up vote 1 down vote accepted

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 $x^{(1)}<x^{(0)}$ and $r_0<B$. You can then recursively divide $x^{(1)}$ by $B$ (note that this will require only a finite number of steps), obtaining $$ x=r_0+B\left(r_1+B\left(r_2+B\left(\dotso +B(r_n)\right)\right)\right) $$ with $0\leq r_i<B$. Therefore $b_i=r_i$ as required.

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.

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