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

I need assistance in solving this one.

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    http://www.mathsisfun.com/binary-decimal-hexadecimal-converter.html is worth a look. It offers decimal, binary and hex conversions. It might help you see the relationships, ex, enter binary .010101010101 and see what this is in decimal.2014-07-08

3 Answers 3

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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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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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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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    @Sid, yes --- just think about what it means to write a number to a base.2014-07-09