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 (https://ncalculators.com/digital-computation/binary-division-calculator.htm) 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?
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$$\frac{\frac14}{1-\frac14}=\frac13.$$