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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?

A.P.
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KawaiKx
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1 Answers1

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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.

A.P.
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  • How to adapt this to real numbers? What happens to the fractorial part if we can't use division theorem? – user5539357 Mar 29 '20 at 08:31
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    @user5539357 You use division of real numbers, the integer part $\lfloor \cdot \rfloor$ instead of the remainder, and the fractional part ${\cdot} = \cdot - \lfloor \cdot \rfloor$ to get the next number to iterate on. Note that with arbitrary real bases $>1$ the resulting sequence of digits is not unique in general, although a canonical one can be found with the construction described here. – A.P. Mar 29 '20 at 11:22
  • Beware This proves only the existence of such a base $B$ rep. Without knowing (proving) the uniqueness of such reps the claim $,b_i = r_i,$ is unjustified. Thus the argument is either incomplete, or incorrect (possibly an instance of the common fallacy: unique algorithmic steps does not imply the result is unique). A simple way to prove uniqueness is to use the Rational Root Test. $\ \ $ – Bill Dubuque Mar 14 '25 at 19:14
  • @BillDubuque Indeed, the argument is sort of incomplete, but only in the sense that I didn't think that proving uniqueness was pertinent to the question, which was specifically about an algorithm for switching between different representations of the same number. I think that you read in $b_i = r_i$ more than I intended. All I meant by that is that the $r_i$ are the coefficients of a representation of the given number in base B. – A.P. Mar 15 '25 at 20:23