How to convert $\pi$ to base 16?

Question

According to this Wikipedia article $\pi$ is approximately 3.243F in base 16 (i.e. hexadecimal).

Can someone explain this? (Note: I understand how to convert an integer to base 16)

Thanks

Answer 1

One way to convert any decimal fraction to base $16$ is as follows (taking $\pi$ as an example).$$\pi=\color{blue}3.141592...$$

Take the whole number part and convert it to base $16$ as usual. In this case $\color{blue}3$ will remain as $3$. So we have so far got $3.14159..._{10}=\color{red}{3...._{16}}$

This now leaves us with $0.141592...$ - Multiply this by our new base to get $$\color{red}{16}\times0.14159...=\color{blue}2.26544...$$Now again convert the whole number part to our new base as usual - in this case the $\color{blue}2$ remains as a $2$. So we have so far got $3.14159..._{10}=\color{red}{3.2..._{16}}$

This now leaves us with $0.26544...$ - Multiply this by our new base to get $$\color{red}{16}\times0.26544...=\color{blue}4.24704...$$Now again convert the whole number part to our new base as usual - in this case the $\color{blue}4$ remains as a $4$. So we have so far got $3.14159..._{10}=\color{red}{3.24..._{16}}$

This now leaves us with $0.24704...$ - Multiply this by our new base to get $$\color{red}{16}\times0.24704...=\color{blue}3.95264...$$Now again convert the whole number part to our new base as usual - in this case the $\color{blue}3$ remains as a $3$. So we have so far got $3.14159..._{10}=\color{red}{3.243..._{16}}$

You can continue this process for as many digits as you require.

Answer 2

Note that: $$10000\pi=31415.92653\dots$$ which means that the decimal version of $\pi$ begins $3.1415\dots$.

Similarly: $$16^4\pi=205887.46145\dots$$ Since $205887$ is $3243F$ in hexadecimal, the hexadecimal version begins $3.243F\dots$.

Answer 3

For the particular base of $16$, there is this remarkable formula: $$\pi=\sum_{n=0}^\infty \left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)\frac{1}{16^n}$$ It allows the computation of any base 16 digit of $\pi$ without the need to compute all the preceding digits.

The discovery of this formula by Bailey, Borwein and Plouffe in 1995 came as a big surprise, as it was conjectured that no such formula can exist.