What are the first 5 digits after the decimal point (technically the hexadecimal point...) when the fraction $\frac{2}{7}$ is written in base $16$?

Answer 1

To convert the fraction $\frac{2}{7}$ into base $16$:

We first need to convert $2$ and $7$ into base $16$:
$2=2_{10}=2_{16}$
$7=7_{10}=7_{16}$

To find the decimal expansion of $\frac{2}{7}$, we can divide $2$ by $7$ (after converting to base $10$).

$2$ divided by $7$ in base $10$ is $0.\overline{285714}$

Converting this recurring decimal into base $16$ (hexadecimal) gives $0.4B216379ECF4$...

So, the first 5 digits after the hexadecimal point are $4B21$.