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Math Constants & Their Histories

This document discusses the history of calculating important mathematical constants like pi, Euler's number, and the imaginary number i. It describes how Babylonian mathematicians first approximated pi as 3 1/8 around 2000 BCE, and how Archimedes later calculated pi to be between 3 1/7 and 3 10/71 around 250 BCE. The document also outlines John Wallis' product formula for pi from 1655, John Machin's practical formula for computing pi in 1706, and the proof that e and pi are transcendental numbers in 1873.

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79 views20 pages

Math Constants & Their Histories

This document discusses the history of calculating important mathematical constants like pi, Euler's number, and the imaginary number i. It describes how Babylonian mathematicians first approximated pi as 3 1/8 around 2000 BCE, and how Archimedes later calculated pi to be between 3 1/7 and 3 10/71 around 250 BCE. The document also outlines John Wallis' product formula for pi from 1655, John Machin's practical formula for computing pi in 1706, and the proof that e and pi are transcendental numbers in 1873.

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© © All Rights Reserved
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KNOWN NUMBERS IN

MATHEMATICS AND THE


HISTORY OF THEIR
CALCULATIONS .

Iga Walczak & Kacper Drozd


AGENDA:

1. Episodes in the calculation of pi.

2. Euler's number.
3. Imaginary number i
4. Golden ratio phi
 By 2000 B.C.E. several cultures we
collectively call the Babylonians
used 3 1/8 as the value of π.

 There is little indication in either


• A LITTLE BIT OF
the Babylonian writings or in
HISTORY the Bible that the values of π
are understood to
be just approximations.
RHIND PAPYRUS


ARCHIMEDES OF
SYRACUSE


https://ohistorii.blogspot.com/2015/03/archimedes-i-wdrozenia-nauk.html
ARCHIMEDES' WORK 
"MEASUREMENT
OF THE CIRCLE."

A picture taken from: http://www.ams.org/publicoutreach/feature-column/fc-2012-02


 In an algebraic formulation,
we say that the area of a circle is πr^2
and its circumference is 2πr.
 These are consistent with Archimedes' claim:
πr^2=(1/2)⋅r⋅(2πr).

 But the ancient Greeks did not have algebra,


and they did not have the notion of a
real number that we do.

 Almost all their `formulas' are in the same


style that this one is - they assert that two
areas are equal.
APPROXIMATION
FOR THE VALUE OF
PI, COUNTED BY
ARCHIMEDES


https://ctsciencecenter.org/blog/archim edes-the-mathematician/
https://pl.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te

FRANÇOIS VIÈTE


PROOF OF VIÈTE'S THEOREM
JOHN WALLIS 

https://en.wikipedia.org/wiki/Wallis_produ ct
JOHN MACHIN'S
FORMULA

A picture taken from: https://en.wikipedia.org/wiki/John_Machin


JOHN MACHIN'S FORMULA

• The arithmetic required to divide by ascending powers of 5 is easily handled, due to the simple
terminating decimal expansions of such fractions.

• Machin now had a practical formula that could compete with, and defeat all previous methods
dependent on multisided polygons.
• In fact, after significant effort, Machin calculated pi to 100 decimal places.
e is proven to be transcendental . It was proven by Charles Hermit using Lindemann-
Weierstrass theorem in 1873.

From a number theory view e is also a normal number.


a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed
uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to
be normal in base b if, for every positive integer n, all possible strings n digits long have density b−n.
BRIEF HISTORY


 https://pl.wikipedia.org/wiki/Pi
 https://www.newworldencyclopedia.org/entry/John_Wallis
 https://en.wikipedia.org/wiki/Wallis_product
 https://ctsciencecenter.org/blog/archimedes-the-mathematician/
 http://personalpages.to.infn.it/~zaninett/pdf/machin.pdf

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