Transcript that at its deepest level, reality is mathematical in nature, that

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, and
5.that all brothers of the order should observe strict loyalty and secrecy.
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Samos
Pythagoras Quotes:
“Numbers rule the Universe”
“Geometry is knowledge of
eternally existent”
“Number is the within of all
things”
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Abstract Mathematics
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?
Powers
Xn 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, Continued
Some mathematical operations are made easier using
powers, for example:
Xn  Xm = Xn+m
therefore 32 = 4  8 = 22  23 = 22+3= 25= 32
Powers of Ten
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
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
stars
in the Universe
Scientific Notation
Any number can be written as a sequence of integers
multiplied by powers of ten. For example
1,234,567 = 1.234567106
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 = 5.495102 (not 103)
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 Figures
The relative importance of the digits in a number
written in scientific notation decrease to the right.
For example, 1.234567106 is very close to
1.234566106, but 2.234567106 is quite different from
1.234567106.
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 1.234106. The number of digits
that we keep is number of significant figures.
1.234567106 has 7 significant figures, but
1.234106 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 Zero
Why is zero important? Because it enables the placevalue 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
Some Simple Geometry
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
=3.1415926535897932384626433832795028841
971693993751....
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
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/Ra
manujan
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
Archimedes: Antiquity’s Greatest Scientist
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The
discovery
Archimedes
was most
proud of
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Spheres
The 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
4
1
example meters cubed,
V  which
R  D is usually denoted as
3
6
3
m , though you might sometimes see it spelled out as
meters cubed. 
3
3
We can also measure the area on the surface of a
sphere, called the surface area (A),
A = 4R2 or A = D2
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