Math ReviewPythagoras of Samos (570-500 B.C) and the Invention of MathematicsPowerPoint PresentationSlide 4PowersPowers, Continued…Powers, ContinuedSlide 8ExampleSlide 10ExamplesSignificant FiguresHow Many Significant Figures are Displayed on Your Calculator?Slide 14Digression on ZeroDigression on Mayan MathematicsSlide 17Slide 18Digression on Slide 20Slide 21SpheresSlide 23Slide 24Slide 25Math ReviewAlong with various other stuffNATS206-224 Jan 2008Pythagoras of Samos (570-500 B.C) and the Invention of MathematicsPythagoras 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 decompressorare needed to see this picture.QuickTime™ and a decompressorare 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 sheep1000 Persian Ships x 100 Persians/ship = 100,000 Persians-Or –2 + 2 = 4100 x 1000 = 100,000Why bother with the sheep and Persians?Abstract MathematicsPowersXn 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/XnExample: 3-2 = 1/32 = 1/9 = 0.1111111Powers, ContinuedSome mathematical operations are made easier using powers, for example:Xn Xm = Xn+mtherefore 32 = 4 8 = 22 23 = 22+3= 25= 32Xn means X multiplied by itself n times10n means 10 multiplied by itself n times10-n means 1 divided by 10nPowers of ten are particularly easy1=100; 10=101; 100=102; 1000=103; 10,000=104Obviously, 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 point1=100; 0.1=10-1; 0.01=10-2; 0.001=10-3; 0.0001=10-4Powers of TenExample•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 UniverseAny number can be written as a sequence of integers multiplied by powers of ten. For example1,234,567 = 1.234567106Notice 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 108Tallest building, 549.5 meters = 5.495102 (not 103)Scientific NotationExamples•How many seconds in 1 year?60 seconds in 1 minute60 minutes in 1 hour24 hours in 1 day365.25 days in 1 yearSec/year = 60x60x24x365.25Significant FiguresThe 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 1.234566106, 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, but1.234106 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 lbsDigression 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 MathematicsThe 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 minutesCircles: 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 byA= 1/4 D2Sometime radius (R) is used in place of diameter. The radius of a circle or sphere is equal to half its diameter: R=D/2Some Simple GeometryDigression on Source Date ValueOld Testament 500 BC 3Archimedes 250 BC 3.1463Tsu Ch’ung Chi 450 AD 355/113Al’Khwarizimi 800 AD 3.1416Ludolph Van Ceulen1600 35 digitsRamanujan 1900 Derived formulaChudnovskys/Ramanujan1990 2 billion digitsProject Gutenberg 1995 1,254,539 digitsExample•How far is it from the north pole to the equator?•Diameter of Earth = 7901 milesQuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.The discovery Archimedes was most proud ofArchimedes: Antiquity’s Greatest ScientistSpheresThe volume (V) of a sphere is equal toV = 4/3 R3 or V = 1/6 D3 We measure volume in units of length
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