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. . .