Pi hiding in prime regularities

Three mysterious Math Objects:

• Pi, 
• Complex Numbers, 
• Prime Numbers

are hiding in circle.

Advertisements

神童背诵圆周率秀心算

5岁儿童背 Pi 600小数位:

Ref:
Math Chant: Pi (22 decimal places)

Pi in the Bible — 1 Kings 7:23

This is the first appearance of

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:

100-digit Pi

One fine day when we reach above 80 years old, if the doctor accuses us of having dementia, then prove the doctor wrong by shocking him with 100-digit Pi memory 🙂

With Chinese single-syllable sound for numbers, better still if can sing it as a song, memorizing 100-digit pi is easy!

Transcendental Number

Transcendental numbers: e, Π, L…

What about ?

Aleksander (Alexis) Osipovich Gelfond (1906-68):

Gelfond-Schneider Theorem

transcendental if
a is algebraic, not 0 or 1
b irrational algebraic number

Examples:

Hilbert Number: (Hilbert Problem proven by Gelfond}

Is log 2 transcendental ?
[log = logarithm Base 10]

Proof:

1) Sufficient to prove log 2 irrational
Assume log 2 rational
log 2= p/q, p and q integers

raise power q

Case 1: p>q

=> False

Case 2: q>p

Left is even :
Right is odd:
=> False

Therefore p,q do not exist,
=> log 2 irrational

Reference: Top 15 Transcendental Numbers: http://sprott.physics.wisc.edu/pickover/trans.html