Binary division: 1/11

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The exercise in the book goes like this: In the decimal system the fraction 1/3 is written as 0.3333... (pure periodic number). What happens in the binary system?

So I've been practising with other divisions in binary and, comparing my result with an online calculator () everything seems to go smoothly. However, when it comes to this particular division the result differs greatly:

1/11 in binary-->

• According to me = 0,01010... This is the logic I follow: 1 divided by 11 is impossible, so I need to add a '0' to the one and a 0 to the quotient as well. Thus, 10/11. Still not possible to divide, hence I add another '0' which turns into 100/11; another 0 to the quotient. NOW I can divide.
• According to the online calculator = 10101010,1

Why is that? Why does the comma come so further away from where I put it?

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2 Answers

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Yes, in binary the fraction $\frac13$ is written as $0.01010101\ldots$ That's so because what $0.01010101\ldots$ means is$$\frac1{2^2}+\frac1{2^4}+\frac1{2^6}+\cdots$$and the sum of this series is$$\frac{\frac14}{1-\frac14}=\frac13.$$

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@LocoVoco Thanks for your discussion, the issue has been now fixed.binary division calculator

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