Math Review

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Math Review

• Along with various other stuff
• NATS206-2
• 24 Jan 2008

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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

• that at its deepest level, reality is
  mathematical in nature,
• that philosophy can be used for spiritual
  purification,
• that the soul can rise to union with the divine,
• that certain symbols have a mystical
  significance, and
• 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 allthings
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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?
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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?
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Powers, Continued
The power need not be an integer. Fractional
Powers Example 21/21.414 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
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Powers, Continued
Some mathematical operations are made easier
using powers, for example Xn ? Xm
Xnm therefore 32 4 ? 8 22 ? 23 223 25
32
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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 1100 10101 100102 1000103
10,000104 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 1100 0.110-1 0.0110-2
0.00110-3 0.000110-4
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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 x10111022 starsin
  the Universe

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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
295,734,1342.95734134 ?108 Tallest building,
549.5 meters 5.495?102 (not 103)
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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

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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. Lets say that we
are lazy and we dont 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.
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How Many Significant Figures are Displayed on
Your Calculator?
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Examples

• Net Weight of People in the USA
• of people in USA 295,734,1342.95734134 ?108
• Average weight of a US Male 185 lbs
• Average weight of a US Female 163 lbs

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Digression on Zero
Why 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.
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Digression on Mayan Mathematics
The ancient Maya were accomplished mathematicians
who developed a number system based on 20
(perhaps they didnt wear shoes).
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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

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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 RD/2
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Digression on ?
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Example

• How far is it from the north pole to the equator?
• Diameter of Earth 7901 miles

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Archimedes Antiquitys Greatest Scientist
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 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 4?R2 or
A ?D2
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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.
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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?

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Example

• What is the area of the Earth?
• Diameter of Earth 12,756 km