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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 $[0, 6]$

J. W. Tanner
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ysmflsd
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1 Answers1

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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 $0.000\,000\,445\,544\,554\ldots$

José Carlos Santos
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