Pi

The following table gives a short list of common approximations used (and misused) for $\pi$, along with their inventor/discoverer.

After the discovery of arctan $x=x-\frac{{\mathit{ x}}^3}{3}+\frac{{\mathit{ x}}^5}{5}-\frac{{\mathit{ x}}^7}{7}+\cdots$, there was a rush of computation, all with slates, chalk, pens, parchment, sticks, sand; no pocket calculators or even slide rules at first.

The most obvious series is if $x=1$; then

$\arctan 1=\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$;

a great little formula but it takes thousands of terms just to get $3$ or $4$ places of accuracy for $\pi$.

Now if you notice that

$\begin{array}{l}\frac{\pi }{4}=2\arctan \left(\frac{1}{3}\right)+\arctan \left(\frac{1}{7}\right)\\ =\frac{2}{3}-\frac{2}{3\cdot 3^3}+\frac{2}{5\cdot 3^5}-\cdots +\frac{1}{7}-\frac{1}{3\cdot 7^3}+\frac{1}{5\cdot 7^5}-\cdots \end{array}$

you do have to deal with two series, but they will "converge" much faster to the exact (irrational) value of pi. Fractions are normal things to try, but since $\pi$ is irrational (not the ratio of two integers), you'll never get it exactly that way.

Lambert proved the irrationality of pi in $1761$. So that rules out $\frac{22}{7}$ and $\frac{355}{113}$.

In $1882$ Lindemann proved that $\pi$ is transcendental, meaning not the root of any polynomial equation. This means $\pi$ squared is not $10$, and $\pi$ cubed is not $31$. But those are some close calls!

These days there are people like the Chudnovsky brothers who calculate to literally billions of places; the computational theory alone is pushing the quality, and not only the size, of the $\pi$ envelope.