1. Math Review Along with various other stuff NATS206-2 24 Jan 2008

2. 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.

3. Samos “Numbers rule the Universe” “Geometry is knowledge of eternally existent” “Number is the within of all things” Pythagoras Quotes:

4. 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

5. Powers X n means X multiplied by itself n times, where n is referred to as the power. Example: 2 2 = 4. Raising a number to the power of two is also called squaring or making a square. Why is this? Example: 2 3 = 8. Raising a number to the power of three is also called cubing or making a cube. Why is this?

6. Powers, Continued… The power need not be an integer. Fractional Powers: Example: 2 1/2 =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/X n Example: 3 -2 = 1/3 2 = 1/9 = 0.1111111

7. Powers, Continued Some mathematical operations are made easier using powers, for example: X n  X m = X n+m therefore 32 = 4  8 = 2 2  2 3 = 2 2+3 = 2 5 = 32

8. X n means X multiplied by itself n times 10 n means 10 multiplied by itself n times 10 -n means 1 divided by 10 n Powers of ten are particularly easy 1=10 0 ; 10=10 1 ; 100=10 2 ; 1000=10 3 ; 10,000=10 4 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=10 0 ; 0.1=10 -1 ; 0.01=10 -2 ; 0.001=10 -3 ; 0.0001=10 -4 Powers of Ten

9. Example There are approximately 100 billion stars in the sky. 1 billion = 1000 million = 10 9 100 billion = 100 x 10 9 =10 2 x 10 9 =10 11 There are at least 100 billion galaxies. So there are at least 10 11 x10 11 =10 22 stars in the Universe

10. Any number can be written as a sequence of integers multiplied by powers of ten. For example 1,234,567 = 1.234567  10 6 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  10 8 Tallest building, 549.5 meters = 5.495  10 2 ( not 10 3) Scientific Notation

11. 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

12. Significant Figures The relative importance of the digits in a number written in scientific notation decrease to the right. For example, 1.234567  10 6 is very close to 1.234566  10 6, but 2.234567  10 6 is quite different from 1.234567  10 6. 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  10 6. The number of digits that we keep is number of significant figures. 1.234567  10 6 has 7 significant figures, but 1.234  10 6 has 4 significant figures.

13. How Many Significant Figures are Displayed on Your Calculator?

14. Examples Net Weight of People in the USA # of people in USA = 295,734,134=2.95734134  10 8 Average weight of a US Male = 185 lbs Average weight of a US Female = 163 lbs

15. 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.

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

17. 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

18. 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  D 2 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

19. Digression on  SourceDateValue Old Testament500 BC3 Archimedes250 BC3.1463 Tsu Ch’ung Chi450 AD355/113 Al’Khwarizimi800 AD3.1416 Ludolph Van Ceulen 160035 digits Ramanujan1900Derived formula Chudnovskys/Ra manujan 19902 billion digits Project Gutenberg19951,254,539 digits

20. Example How far is it from the north pole to the equator? Diameter of Earth = 7901 miles

21. The discovery Archimedes was most proud of Archimedes: Antiquity’s Greatest Scientist

22. Spheres The volume (V) of a sphere is equal to V = 4/3  R 3 or V = 1/6  D 3 We measure volume in units of length cubed, for example meters cubed, which is usually denoted as m 3, 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  R 2 or A =  D 2

23. 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.

24. 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?

25. Example What is the area of the Earth? Diameter of Earth = 12,756 km