Binary form of a number

Question

I'm a little confused about the following example given by my textbook on have to convert $\frac{2}{3}$ to its binary form

To determine the binary representation we write

$\frac{2}{3}=(0.a_1a_2a_3...)_2$

We multiply by 2 to obtain

$\frac{4}{3}=(a_1.a_2a_3...)_2$

Therefore we get $a_1=1$ by taking the integer part of both sides. Subtracting 1 from both sides, we have

$\frac{1}{3}=(0.a_2a_3a_4...)_2$

Repeating the previous steps , we eventually arrive at

$\frac{2}{3}=(0.1010....)_2$

Now I get why $a_1=1$ by why does $a_2=0$?

Answer 1

From $$\frac{1}{3}=(0.a_2a_3a_4...)_2$$ we multiply by $2$ to obtain $$\frac23=(a_2.a_3a_4\ldots)_2$$

Since $2/3<0$, $a_2$ is $0$.