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

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!

• Sorry but your avatar is too adorable. – David Richerby Jan 11 at 19:19

Here is a hint by example.

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

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 $$\frac {23}6$$?