Bailey-Borwein-Plouffe Formular for Pi

With the BBP formular, we can find the 10th billion digit of Pi without calculting all the digits before it.

e^pi vs. pi^e

Prove:

$\boxed{e^{\pi} > \pi^{e}}$

Pi hiding in prime regularities

Three mysterious Math Objects:

• Pi,
• Complex Numbers,
• Prime Numbers

are hiding in circle.

5岁儿童背 Pi 600小数位:

Pi in the Bible — 1 Kings 7:23

This is the first appearance of
$\boxed {\pi = 3 }$
in any human’s text in history.

45 feet / 15 feet = 3

Transcendental Numbers: e, pi

The French Mathematician and Physicist Joseph Fourier proved
e is irrational,

Another French mathematician Charles Hermite went further: e belongs to another mathematical world:
e is transcendental.

Hermite’s German student Lindermann followed the same method, proved:
pi is also transcendental.

Machin’s Pi

1706 John Machin developed a formula for calculating Pi to an arbitrary number of decimal places:

$\boxed{\displaystyle \pi = \text {4 [4.arccot(5) - arccot(239)]} }$

$\boxed {\displaystyle \text{arccot (x)} = \frac{1}{x} - \frac {1}{3{x^{3}}} + \frac {1}{5{x^{5}}}-\frac {1}{7{x^{7}}} + \cdots }$