How to transform a decimal of $\frac{23}{6}$ into binary?

$\frac{23}{6}$ is equal to $3.83333\bar{33}$, where the digit $3$s are repeated forever.

I do know how to translate finite decimal into binary, but is there any procedure to translate such Infinite decimal like $\frac{23}{6}$ into binary?

Any hint is appreciated!

  • $\begingroup$ Sorry but your avatar is too adorable. $\endgroup$ – David Richerby Jan 11 at 19:19
  • $\begingroup$ @DavidRicherby lol thank you :) $\endgroup$ – OOD Waterball Jan 12 at 4:30

Here is a hint by example.

$\frac{23}6 > 2$
$\frac{23}6-2 = \frac {11}6 > 1$
$\frac{11}6-1 = \frac {5}6 > \frac 12$
$\frac{5}6-\frac12 = \frac {1}3 > \frac 14$
$\frac{1}3-\frac14 = \frac {1}{12} > \frac 1{16}$

So $\frac{23}6=2+1 +\frac12+\frac14+\frac1{16}+\cdots=(11.1101\cdots)_2$

Exercise. How to determine the period of the fraction part of the binary expansion of $\frac {23}6$?


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