Why is $\frac {4\sqrt{40}}{\log_{10}{40}} - 4\sqrt{10}$ so close to $\pi$?

Question

Why is $$\frac {4\sqrt{40}}{\log_{10}{40}} - 4\sqrt{10}\approx 3.1419$$ so close to $\pi$?

Answer 1

Just the same way as $\sqrt{10}, \frac{22}{7}$ are close to $\pi$.

Answer 2

You know $\pi\approx 3.141592$. The interval $[a,b]$ where $a=3.141591$ and $b=3.141593$ contains uncountable many numbers (rational, algebraic and trascendental ones) close to $\pi$. Anyway, to find out closed forms for $\pi$ is a matter of contemporary research and, for instance the number$$\frac{\ln(640320^3+744)}{\sqrt{163}}$$ gives $30$ exact decimal digits of approximation. The number will be much more valuable the greater the approximation be and the number above is not easy to obtain.

Answer 3

By combining up to 15 mathematical symbols from the dozens available, you can make trillions of numbers. Some of them will be close to $\pi$, including $3.1416$, which requires only 6 symbols.