1

πIt is quite curious that π is related to probability. The probability that two natural numbers selected at random will be relatively prime is

2

6

This is quite astonishing since π is derived from a geometric setting.

2

Pertinent Dates• ca. 546 BC Thales

• ca. 585 – 500 B.C. Pythagoras

•485 – 410 B.C. Proclus

• 469 - 399 B.C. Socrates

• 455 – 385 B.C. Aristophanes

• 427 – 347 B.C. Plato

• 384 – 322 B.C. Aristotle

• ca. 300 B.C. Euclid

3

3 – Pythagorean Mathematics

The student will learn about

Greek mathematics before the time of Alexander the Great.

4

Cultural ConnectionThe Philosophers of the Agora

Hellenic Greece – ca. 800 – 336 B.C.

Student led discussion.

5

§3-1 Birth of Demonstrative Mathematics

Student Discussion.

6

§3-1 Birth of Demonstrative Mathematics

Statement Reason

Theorem: Vertical angles formed by two intersecting lines are equal.

Proof: < 1 = < 3

< 1 + < 2 = 180 Straight line

12

34

< 2 + < 3 = 180 Straight line

< 1 + < 2 = < 2 + < 3 Substitution

< 1 = < 3 Subtraction

QED w 5

7

§3-2 Pythagoras and the Pythagoreans

Student Discussion.

8

§3-3 Pythagorean Arithmetic 1

Student Discussion.

9

§3-3 Pythagorean Arithmetic 2

Perfect numbers equal the sum of their proper divisors. 6

Abundant numbers exceed the sum of their proper divisors. 12

Deficient numbers are less than the sum of their proper divisors. 8

10

§3-3 Pythagorean Arithmetic 3Euclid proved that if 2n – 1 is prime then

2n – 1(2n – 1) is perfect.

n 2n 2n – 1 2n – 1 2n – 1(2n – 1)

1 2 1 1 1

2 4 3 2 6

3 8 7 4 28

4 16 15 8 240

5 32 31 16 496

6 64 63 32 2016

7 128 127 64 8128

11

§3-3 Pythagorean Arithmetic 3Figurative numbers

Triangular numbers

Tn = =

n

1ii n

n + 1

Square numbers

1 3 6

1 4 9

Sn = n2 = n (n + 1) / 2 + n (n - 1) / 2 = Tn + Tn - 1

Tn = = n (n + 1) / 2

12

§3-4 Pythagorean Theorem 1

Student Discussion.

13

14

§3-4 Pythagorean Theorem 2Pythagorean dissection proof.

a

a

a

a

b

b

bb

cc

c

c

c

=

a

a

a

a

b

b

b

b

15

§3-4 Pythagorean Theorem 3Bhaskara’s dissection proof.

a

a

a

bb

b

b

a

c

c

c

c

c2 = 4 · ½ · a · b + (b – a)2

16

§3-4 Pythagorean Theorem 4Garfield’s dissection proof.

a

b

b

ac

c

½ (a + b) · (a + b) = 2 · ½ · a · b + ½ · c2

17

http://www.usna.edu/MathDept/mdm/pyth.html

18

§3-5 Irrational Magnitudes 1

Student Discussion.

19

§3-5 Irrational Magnitudes 2Geometric interpretation of 2/3.

0 1

2/3

20

§3-5 Irrational Magnitudes 3

as the diagonal of a unit square.

Proof that is irrational. (Aristotle 384 – 322 B.C.)

Assume is rational. I.e. = a/b & a and b are relatively prime.

2

2

2 2

Then 2 = a2/b2 and a2 = 2 b2 and hence a is even.

Let a = 2k since it is even and then

4k2 = 2 b2 and hence b is even.

a contradiction hence is not rational.2

21

§3-6 Algebraic Identities 1

Student Discussion.

22

§3-6 Algebraic Identities 2(a + b)2 = a2 + 2ab + b2.

a

a

a

a

b

b

b b

a + b

23

a 2

§3-6 Algebraic Identities 3(a - b)2 = a2 - 2ab + b2.

a - b

a

b

b

b b

a - b

a - b a - b1 2

3 4a 2 – aba 2 – ab – aba 2 – ab – ab + b 2

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§3-7 Geometric Solutions of Equations 1

Student Discussion.

25

§3-7 Geometric Solutions of Equations 2

Linear equations a x = b c

ab

c

x

bcaxorx

c

b

a

26

§3-7 Geometric Solutions of Equations 3

Quadratic equations x2 = a b.

a b

x

abxorb

x

x

a 2

27

§3-8 Transformation of Areas 1

Student Discussion.

28

§3-8 Transformation of Areas 1

Construct a square equal in area to a given polygon.

Given ABCDE

Construct BR AC with R on DC

Area ABC = area ARC

Hence area of ABCDE = ARDE

A B

C

D

E

R

29

§3-8 Transformation of Areas 2

Construct a square equal in area to a given polygon.

Given ARDE = ABCDE

Construct RS AD with S on ED

Area ARD = area ASD

Hence area of ABCDE = ARDE = ASE A

S

C

D

E

R

Make it a square!

30

§3-9 The Regular Solids 1

Student Discussion.

31

§3-9 The Regular Solids 2

• Tetrahedron

• Hexahedron

• Octahedron

• Dodecahedron

• Icosahedron

Show models

Show imbedded model.

32

§3-10 Postulational Thinking

Student Discussion.

33

Assignment

Read Chapter 4.

Outline of Paper 1 due on Monday!