TITLE: How to prove that when we change numeral system same number becomes shorter
QUESTION [3 upvotes]: Let's start with an example:

We have a binary number $1100_2$ 
In decimal it will be equal to $12_{10}$
In hexadecimal it will be equal to $C_{16}$

I'm trying to find a proof that for any positive integer number if we will change numerical system from $a$ to $b$, where $a > b$, the following will be always true:
Number $X_a$ will always have less or same amount of characters as the same number $X_b$

REPLY [3 votes]: If you represent the number $n$ in the base $a$, you get$$a_0+a_1a+\cdots+a_ka^k;\tag1$$in base $b$, you get$$b_0+b_1b+\cdots+b_lb^l.\tag2$$ Each $a_j$ belongs to $\{0,1,\ldots,a-1\}$ and each $b_j$ belongs to $\{0,1,\ldots,b-1\}$. If $l>k$, then$$(2)\geqslant b^l>a^l>(1),$$which is impossible, since both $(1)$ and $(2)$ are equal to $n$.
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