To convert the fraction $\frac{2}{3}$ to base 16, we need to multiply the numerator and denominator by $16$ to get integers in base 16. Then, we have $\frac{32}{48}$.
Next, we can simplify this by dividing both the numerator and denominator by $16$ to get $\frac{2}{3}$. However, in base 16, we need to express the numerator as a single-digit number. The closest multiple of 16 to $3$ is $16 \cdot 1 = 16$. So, we can write $\frac{32}{48}$ as $\frac{16 \cdot 2}{16 \cdot 3} = \frac{2}{3}$.
The first digit after the hexadecimal point in base 16 is $1$, which is equivalent to $1$ in base 10. The next four digits are $999A$. Therefore, the first $5$ digits after the decimal point when $\frac{2}{3}$ is written in base $16$ are $1999A$.
What are the first $5$ digits after the decimal point (technically the hexadecimal point...) when the fraction $\frac{2}{3}$ is written in base $16$?
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