Math Review

Along with various other stuff

NATS206-2

24 Jan 2008

Pythagoras of Samos (570-500 B.C) and the Invention of Mathematics

Pythagoras founded a philosophical and religious school in Croton (Italy) that had enormous influence. Members of the society were known as mathematikoi. They lived a monk-like existence, had no personal possessions and were vegetarians. The society included both men and women. The beliefs that the Pythagoreans held were:

1.that at its deepest level, reality is mathematical in nature,2.that philosophy can be used for spiritual purification,3.that the soul can rise to union with the divine,4.that certain symbols have a mystical significance, and5.that all brothers of the order should observe strict loyalty and secrecy.

QuickTime™ and a decompressor

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QuickTime™ and a decompressor

are needed to see this picture.

Samos

“Numbers rule the Universe”“Geometry is knowledge of eternally existent”“Number is the within of allthings”

Pythagoras Quotes:

2 sheep + 2 sheep = 4 sheep

1000 Persian Ships x 100 Persians/ship = 100,000 Persians

-Or –

2 + 2 = 4

100 x 1000 = 100,000

Why bother with the sheep and Persians?

Abstract Mathematics

PowersXn means X multiplied by itself n times, where n is referred to as the power.

Example: 22 = 4. Raising a number to the power of two is also called squaring or making a square. Why is this?

Example: 23 = 8. Raising a number to the power of three is also called cubing or making a cube. Why is this?

Powers, Continued…The power need not be an integer.

Fractional Powers:

Example: 21/2=1.414 Raising a number to the power of 1/2 is also called taking the square root.

Negative Powers:

Raising a number to a negative power is the same as dividing 1 by the number to the positive power, I.e.

X-n = 1/Xn

Example: 3-2 = 1/32 = 1/9 = 0.1111111

Powers, ContinuedSome mathematical operations are made easier using powers, for example:

Xn Xm = Xn+m

therefore 32 = 4 8 = 22 23 = 22+3= 25= 32

Xn means X multiplied by itself n times

10n means 10 multiplied by itself n times

10-n means 1 divided by 10n

Powers of ten are particularly easy

1=100; 10=101; 100=102; 1000=103; 10,000=104

Obviously, the exponent counts the number of zeros.

For negative powers of ten, the exponent counts the number of places to the right of the decimal point

1=100; 0.1=10-1; 0.01=10-2; 0.001=10-3; 0.0001=10-4

Powers of Ten

Example

• There are approximately 100 billion stars in the sky.

• 1 billion = 1000 million = 109

• 100 billion = 100 x 109 =102 x 109 =1011

• There are at least 100 billion galaxies.• So there are at least 1011 x1011=1022

starsin the Universe

Any number can be written as a sequence of integers multiplied by powers of ten. For example

1,234,567 = 1.234567106

Notice that on the left there are 6 places after the 1 and on the right ten is raised to the power of 6.

Examples:

# of people in USA = 295,734,134=2.95734134 108

Tallest building, 549.5 meters = 5.495102 (not 103)

Scientific Notation

Examples

• How many seconds in 1 year?

60 seconds in 1 minute

60 minutes in 1 hour

24 hours in 1 day

365.25 days in 1 year

Sec/year = 60x60x24x365.25

Significant FiguresThe relative importance of the digits in a number written in scientific notation decrease to the right.

For example, 1.234567106 is very close to 1.234566106, but 2.234567106 is quite different from 1.234567106.

Let’s say that we are lazy and we don’t want to write down all those digits. We can transmit most of the information by writing 1.234106. The number of digits that we keep is number of significant figures.

1.234567106 has 7 significant figures, but

1.234106 has 4 significant figures.

How Many Significant Figures are Displayed on Your

Calculator?

Examples

• Net Weight of People in the USA• # of people in USA = 295,734,134=2.95734134 108

• Average weight of a US Male = 185 lbs• Average weight of a US Female = 163 lbs

Digression on ZeroWhy is zero important? Because it enables the place-value number system just described. It is difficult to deal with large numbers without zero.

Zero was first used in ancient Babylon (modern Iraq) in the 3rd century BC.

Our use of zero comes from India through the Islamic world and China. The word zero comes from the arabic sifr; the symbol from China. Zero seems to have been invented in India in the 5th century AD, but whether this was independent of the Babylonians is debated.

Independently, Mayan mathematicians in the 3rd century AD developed a place-value number system with zero, but based on 20 rather than ten.

Digression on Mayan Mathematics

The ancient Maya were accomplished mathematicians who developed a number system based on 20 (perhaps they didn’t wear shoes).

Examples

• What fraction of your life is this class occupying?

• Average lifespan for males in USA = 76.23 years

• Average lifespan for females in USA = 78.7 years

• Average length of NATS206 class = 1 hour and 15 minutes

Circles:

The ratio of the circumference of a circle (C) to the diameter (D) is called (‘pi’), C/D= . The quantity is the same for all circles

=3.1415926535897932384626433832795028841971693993751....

The area (A) of a circle is related to the diameter by

A= 1/4 D2

Sometime radius (R) is used in place of diameter. The radius of a circle or sphere is equal to half its diameter: R=D/2

Some Simple Geometry

Digression on Source Date Value

Old Testament 500 BC 3

Archimedes 250 BC 3.1463

Tsu Ch’ung Chi 450 AD 355/113

Al’Khwarizimi 800 AD 3.1416

Ludolph Van Ceulen

1600 35 digits

Ramanujan 1900 Derived formula

Chudnovskys/Ramanujan

1990 2 billion digits

Project Gutenberg 1995 1,254,539 digits

Example

• How far is it from the north pole to the equator?

• Diameter of Earth = 7901 miles

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are needed to see this picture.

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are needed to see this picture.

The discovery Archimedes was most proud of

Archimedes: Antiquity’s Greatest Scientist

SpheresThe volume (V) of a sphere is equal to

V = 4/3 R3 or V = 1/6 D3

We measure volume in units of length cubed, for example meters cubed, which is usually denoted as m3, though you might sometimes see it spelled out as meters cubed.

We can also measure the area on the surface of a sphere, called the surface area (A),

A = 4R2 or A = D2

€

V =4

3πR3 =

1

6πD3

Visualize taking each little segment in this drawing, laying it flat, measuring its area, and adding them all together. This would give you the surface area.

Examples

• What is the area of a room has dimensions of 15’ x 20’?

• What is the area of a room in square feet if the dimensions are 3 yards by 4 yards?

• What are the dimensions of a square room with an equal area?

Example

• What is the area of the Earth?

• Diameter of Earth = 12,756 km