# Which base of numerical system have $\frac 15 = 0.33333\ldots$?

Which base of numerical system have $\frac{1}{5} = 0.33333\ldots$?

I need assistance in solving this one.

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## 3 Answers

Hint: if we multiply $0.33333\ldots$ by $5$ then we get $0.(15)(15)(15)(15)(15)\ldots$. Compare that to what happens when we multiply the same by $3$: $0.99999\ldots$, and its interpretation in decimal.

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If we are working in base $b$ (we must have $b\gt3$), then $0.3333\ldots$ is $$0.3333\ldots = \frac{3}{b} + \frac{3}{b^2} + \frac{3}{b^3}+\cdots$$ Since $$\sum_{n=1}^{\infty}\frac{3}{b^n} = \frac{3}{b}\sum_{n=0}^{\infty}\frac{1}{b^n} = \frac{3}{b}\left(\frac{1}{1-\frac{1}{b}}\right) =\frac{3}{b-1},$$ then...

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A reworking of Arturo's answer: let $x=0.333\dots$, let the base be $b$, then $$bx=3.333\dots$$ so $bx-x=(3.333\dots)-(0.333\dots)$ and you can take it from there.

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That should be $bx-x$, not $bx-b$. –  jwodder Oct 30 '11 at 16:34
@jwodder correction made, thanks very much. –  Gerry Myerson Oct 30 '11 at 21:44
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