Convert to different number base?

Question

If I have the number:

$$ 0.67 \cdot 7^{-6}$$ in base 10, what will the converted value in base $7$ be?

I am finding it difficult in understand the logic of conversion, can someone give any guidance on how it works? I believe the value cant be the same since the digits will range in $[0, 6]$

Answer 1

We have:

• $67\times7=\color{red}4\times100+69$;
• $69\times7=\color{red}4\times100+83$;
• $83\times7=\color{red}5\times100+81$;
• $81\times7=\color{red}5\times100+67$.

Since the remainder $67$ had already occurred, things now enter into a cycle, and therefore $0.67$ is represented in base $7$ as $0.445\,544\,554\,455\ldots$ and therefore $0.67\times7^{-6}$ is represented by $0.000\,000\,445\,544\,554\ldots$