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There is an article called “The Base-Conversion method:Why does it work?” (The Base-Conversion method:Why does it work?),which states that:

Each time you divide, you're asking "Does the original number contain a multiple of this power of two?", and the remainder is either telling you "yes" (with a "0") or "no" (with a "1")”.

I have checked it on many examples and it really works,but I can not understand why. Could you please explain.

Thanks in advance.

user641597
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  • You may find an answer here: https://math.stackexchange.com/questions/3017323/is-there-an-easy-way-to-see-that-binary-expansion-is-unique/3017368#3017368 – Ethan Bolker May 17 '19 at 20:21

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When you write a number in base $2$ the non zero numbers are remainders in dividing by powers of two. For example $15$ is written as $1111$ that means if you divide it by two the remainder is one . $$15=2\times 7+1$$ Now if you divide $7$ by two the remainder is $1$ and so forth.

Mohammad Riazi-Kermani
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