Post on 26-Apr-2020
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
The First Great TheoremThe History of Mathematics, Part 4
Chuck Garner, Ph.D.
Department of Mathematics
Rockdale Magnet School for Science and Technology
January 17, 2014
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Outline
The Pythagorean Theorem before PythagorasBabylonian
Chinese
Egyptian
Pythagoras and the Pythagoreans
Extensions and Generalizations
Homework
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Outline
The Pythagorean Theorem before PythagorasBabylonian
Chinese
Egyptian
Pythagoras and the Pythagoreans
Extensions and Generalizations
Homework
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
BabylonianClay tablet (Plimpton 322) from Babylon, c.2000 bc
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
BabylonianClay tablet from Babylon, c.2000 bc
Second column: shorter leg
Third column: hypotenuse
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
ChineseCalled the “gougu” rule
1600 ad copy of a book wri�en in 1500 bc
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Egyptian
I Knew special right triangles
I Used ropes with knots equally spaced apart
I No formal rules for finding Pythagorean triples
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Outline
The Pythagorean Theorem before PythagorasBabylonian
Chinese
Egyptian
Pythagoras and the Pythagoreans
Extensions and Generalizations
Homework
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagoras
Pythagoras of Samos
570 bc-475 bc
“Do not say a li�le in many words but a great deal in afew.”
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagoras
I Possibly studied under Thales
I Traveled to Egypt and possibly India
I Founded a school in Crotona
I Possibly was the first to prove the Pythagorean
Theorem
I School developed elaborate rites and beliefs
I All discoveries made by students were credited to
Pythagoras
I Political pressure forced Pythagoras to flee
I Died in Metapontum
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
I Whole numbers rule the universe in all ways
I “Divine Number” bestowed blessings
I School emblem was a 5-pointed star
I Women treated as equals
I Became quasi-religious with multiple
“commandments”
I Abstain from beans
I Do not pick up what has fallen
I Do not break bread
I Do not look in a mirror beside a light
I Do not stir the fire with iron
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
Number beliefs
1. The source of all things; the monad
2. The divided being; Janus; feminine
3. Completion and heaven; masculine
4. Perfection; justice and earth
5. Marriage of heaven and earth
6. Luck
7. Cosmic
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
Number beliefs
1. The source of all things; the monad
2. The divided being; Janus; feminine
3. Completion and heaven; masculine
4. Perfection; justice and earth
5. Marriage of heaven and earth
6. Luck
7. Cosmic
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
Number beliefs
1. The source of all things; the monad
2. The divided being; Janus; feminine
3. Completion and heaven; masculine
4. Perfection; justice and earth
5. Marriage of heaven and earth
6. Luck
7. Cosmic
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
Number beliefs
1. The source of all things; the monad
2. The divided being; Janus; feminine
3. Completion and heaven; masculine
4. Perfection; justice and earth
5. Marriage of heaven and earth
6. Luck
7. Cosmic
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
Number beliefs
1. The source of all things; the monad
2. The divided being; Janus; feminine
3. Completion and heaven; masculine
4. Perfection; justice and earth
5. Marriage of heaven and earth
6. Luck
7. Cosmic
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
Number beliefs
1. The source of all things; the monad
2. The divided being; Janus; feminine
3. Completion and heaven; masculine
4. Perfection; justice and earth
5. Marriage of heaven and earth
6. Luck
7. Cosmic
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Pythagorean School
Number beliefs
1. The source of all things; the monad
2. The divided being; Janus; feminine
3. Completion and heaven; masculine
4. Perfection; justice and earth
5. Marriage of heaven and earth
6. Luck
7. Cosmic
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Outline
The Pythagorean Theorem before PythagorasBabylonian
Chinese
Egyptian
Pythagoras and the Pythagoreans
Extensions and Generalizations
Homework
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
A Dissection Proof
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
A Shearing Proof
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Eulcid’s Proof
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
A General Proof
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
An Extension
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Another Extension
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Outline
The Pythagorean Theorem before PythagorasBabylonian
Chinese
Egyptian
Pythagoras and the Pythagoreans
Extensions and Generalizations
Homework
Theorem
Garner
The PythagoreanTheorem beforePythagorasBabylonian
Chinese
Egyptian
Pythagoras andthe Pythagoreans
Extensions andGeneralizations
Homework
Homework
I A recent theory behind the use of Plimpton 322;
Reader, 1.E3
I More on the Pythagorean Theorem;
Math Through the Ages, Sketch 12
Next: The First Crisis. . .