2013 06 04 - MAT750 Notes

Exam Thurs 6/6/13 6:30pm - 9:00pm, Bring Calculator, Coverage Ends, here _________.

Greeks

1500 BCE	The idea of "Greeks" has already spread... a sense of Greekness
1200 BCE	The Fall of Troy
1200 - 800 BCE	The Dark Ages
800 BCE	Homer and Hesiod
	This is the beginning of a Common Culture
7^th Century BCE	Greek Mercenaries in Egypt
510 BCE	When Athens becomes Democratic - They replaced the single-person government with a group and voting.
	The Golden Age of (Mainland) Greece
479 - 431 BCE	The Leader of Athens is Pericles?
431 - 404 BCE	The Athens Sparta War
399 BCE	The death of Socrates
367 BCE	Plato (student of Soc.) Flourished
322 BCE	Aristotle passed (student of Plato)
323 BCE	Alexander the Great passed (student of Aristotle). He gave a Universal State and there was Multi-polarity in Greece
	After Alexander Passes there's Hellenism - language and math
	Successor States - Such as Egypt - a cultural backwater - (Greek and had Alexandria and it's Library).
≈ 300 BCE	Euclid was a math tutor to the king's children. "There is no royal road to Geometry."
	The original textbook on Geometry, Euclid's Elements.
212 BCE	Archimedes is Murdered
31 BCE	Cleopatra Passed - the End of Greek Egypt, and Greek learning for that matter.

Early Greek Math

Had 3 centers

Ionia (Turkish coast)
South Italy/Sicily (Syracuse)
Greece (the poorest of the 3)

Early Math (Greek)

The emphasis was on natural numbers

Geometry - geo(land) metry(measure)

Arithmetic - left to the slaves

The Greek system for numbers was awful... they used their letters α β ε etc.

They were great at commerce

They developed Logic and Proof... Deductive Reasoning

The Greeks offered us the earliest NAMED mathematicians

585 BCE - Thales Flourished - He predicted a solar eclipse in Lydia (in Ionia where Croesus was)

He was a philosopher and geometer... he developed constructions as we know it: Straight Edge and Compass

The earliest mathematician (that we know) who talked about congruent triangles

Gave CPCTC (Congruent Parts of Congruent Triangles are Congruent)

Invented the idea of proof (it was called Theoromata which means "Observations")

He gave us ASA (Angle Side Angle)

He gave us the properties of vertical angles

Studied Isosceles - Iso (equal) sceles (side) - Triangles and proved that they have 2 equal angles

Thales Theorem (not the easiest thing)

Figure 1: Thale's Theorem

Terminology such as: Angle, Polygon, etc goes back to Thales

525 BCE - Pythagoras Flourished in South Italy Pythos(snake) gora(salesman)

Had a whole school of disciples, they were itinerant school (they traveled)

One of his sayings, "All is number" everything in the world can be explained by numbers. This is the start of a godless science.

Numbers are Divine and Numerology (the inner meaning of numbers), Numerolotry (worshiping numbers).

He contributed a lot of number theory, eg. prime numbers

He gave us proof of triangle theorems (like, yes, the Pythagorean Theorem)

He talked about proportions, like Pythagoras' theory of proportions.

Example:

You cannot in general trisect an angle using constructions

As a minor, try John's problem

Pythagorean Proportions

Declare lengths A,B,C,D are in proportion

Written A:B = C:D if:

Figure 2: A:B as C:D or A/B = C/D or AD = BC

A is 1^st rectangle's height, B is 2^nd rectangle's base, C is 2^nd rectangle's height, and D is the 1^st rectangle's base

(The 2 rectangles should have the same area)

AD=BC came from A/B = C/D

Example: 3.5, 4.67, 1.5, 2

Figure 3: Example of a Pythagorean Proportion 3.5 ÷ 4.67 = 1.5 ÷ 2

Left Height / Right Base = Right Height / Left Base

Golden Mean/Section

Given a segment, where to cut it so that the segment → longer piece is to the longer piece : shorter piece

L(L-x)=x • x

L² - Lx = x²

0 = x² + Lx - L²

D = L² - 4(1)(-L²)

L² + 4L² = 5 L²

> 0

(y+w)² + L(y+w) - L² = 0

y² + 2wy + w² + Ly + Lw = L²

y² + y(2w + L) = -(w² + Lw - L²)

y² = -[(L/2)² + L(-L/2)-L²]

y² = -[L²/4 - L²/2 -L²]

y² = 5/4 L²

y=±L √(5)/2

x = ± L √(5)/2 - L/2

Because Length cannot be < 0, just use the +

x = L/2(√(5) - 1) which is > 0 and > L/2

x = L((√(5) - 1)/2) = L Φ The Golden Ratio

Replication using Golden Sections

Figure 4: A recursive and geometric method of producing
Golden Sections.

(2x-1)

Golden?

x/(1-x) = (1-x)/(2x-1)

x ( 2 x - 1)=( 1-x )²

2x² - x = 1-2x+x²

x² + x - 1 = 0 ⇒ x = Φ

Ask: L is the (1-x) > 2x-1?

True iff: 1 > 3x - 1

2 > 3x

2 > 3(( √(5) -1 )/2)

4 > 3√(5) - 3

7 > 3√(5)

49 > 45

1/Φ - 1 = Φ

1 - Φ = Φ²

Φ² + Φ - 1 = 0

Fractals

Figure 5: This Fractal "Monster" is created by recursively
bisecting the sides of an equilateral triangle and connecting
the midpoints.

Golden Pentagon

Figure 6: The Pentagon Naturally has Φ as a ratio of bisectors.
(2.17963)/(2.17963+3.52671) = 0.618... = Φ

Pythagorean Means (10 of them!)

Arithmetic Mean: The desired mean between 2 unequal segments the longer and shorter segments satisfies the following proportion (lesser length=b, bigger length=a) [the excess of the mean over the shorter side] is to the [excess of the longer side over the mean], as the [excess of the mean over the shorter side] is to the [excess of the mean over the shorter side]

x-b:a-x = x-b:x-b ⇒ (x-b)/(a-x) = (x-b)/(x-b) = 1 = x-b = a-x ... 2x=a+b ... x=(a+b)/2

Geometric Mean: The Longer side is to the mean as the mean is to the shorter side

a:x = x:b → ab = x² → x = √(ab) (squaring the rectangle)

a/x = x/b → ab = x² → x = √(ab)

You cannot square a circle (using construction) proved in 18th and 19th centuries nor a shoebox (rectangular parallelepiped)

Ex: 3 & 7 √(21)

THEOREM: 0<a,0<b,a≠b ⇒ Geom. mean < arithmetic mean

PROOF: √(b) - √(a)

(√(b) - √(a))² = b+a - 2√(ab) < (a+b)

√ (ab) < (a+b)/2

Harmonic Mean:

Johnny walked to school with velocity V₁, and runs home with velocity V₂

What is his average velocity for the round trip? It's NOT (V₁+V₂)/2

D=Distance to school

We want V = (total distance)/(total time) = (2D)/((D/V₁)+(D/V₂))

= 2/((1/V₁)+(1/V₂)) = 1/( ((1/V₁) + (1/V₂))/2 )

It's the reciprocal of the average of the reciprocals of the velocities: x^H

These three means suggest that the harmonic mean < geometric mean < arithmetic mean

On a side note about Sir Isaac Newton: Before Newton People could take the derivative and find the area under a curve; Newton said they're the same thing! He unified all this stuff!