Math 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 ofeternally 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 MathematicsPowersXnmeans 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 isthis?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= 32Xnmeans X multiplied by itself n times10nmeans 10 multiplied by itself n times10-nmeans 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 = 1.234567×106Notice 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 = 5.495×102 (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, 1.234567×106is very close to1.234566×106, but 2.234567×106is 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. 1.234567×106has 7 significant figures, but1.234×106 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 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 π (‘pi’), C/D= π. The quantity is the same for all circlesπ=3.1415926535897932384626433832795028841971693993751....The area (A) of a circle is related to the diameter byA= 1/4 πD2Sometime 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 π1,254,539 digits1995Project Gutenberg2 billion digits1990Chudnovskys/RamanujanDerived formula1900Ramanujan35 digits1600Ludolph Van Ceulen3.1416800 ADAl’Khwarizimi355/113450 ADTsu Ch’ung Chi 3.1463250 BCArchimedes3500 BCOld TestamentValueDateSourceExample• 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 πR3or V = 1/6 πD3We 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 = 4πR2or A = πD2V =43πR3=16πD3Visualize 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
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