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Why Logs? From Calculating to Calculus

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John Napier (1550-1617) Scottish mathematician, physicist, astronomer/astrologer Scottish mathematician, physicist, astronomer/astrologer 8th Laird (baron) of Merchistoun 8th Laird (baron) of Merchistoun Famous for inventing logarithms Famous for inventing logarithms Before digital computers, logarithms were vital for computation, at a time when computers were people Before digital computers, logarithms were vital for computation, at a time when computers were people Slide rules are hand computers based on logarithms Slide rules are hand computers based on logarithms Slide rule image downloaded 5- 11-10 from http://en.wikipedia.org/wiki/File :Pocket_slide_rule.jpg

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Tycho Brahe (1546-1601) Born at Knutstorp Castle in Denmark Born at Knutstorp Castle in Denmark Meticulous observer of the stars and planets Meticulous observer of the stars and planets Led the way to proving that the earth revolves around the sun Led the way to proving that the earth revolves around the sun Lived on the Island of Hven Lived on the Island of Hven Lost part of his nose in a duel Lost part of his nose in a duel

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Island of Hven Tycho Brahes Playground Built for Brahe by the King of Denmark at great expense Built for Brahe by the King of Denmark at great expense Active observatory from 1576-1580 Active observatory from 1576-1580 Hosted wild and crazy parties Hosted wild and crazy parties The island had its own zoo The island had its own zoo

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Dr. John Craig (? – 1620) In 1590 Dr. Craig was travelling with James VI of Scotland when he was shipwrecked at Hven In 1590 Dr. Craig was travelling with James VI of Scotland when he was shipwrecked at Hven The incident may have inspired Shakespeares The Tempest The incident may have inspired Shakespeares The Tempest Dr. Craig met Tycho Brahe and learned about the astronomers problems with multiplication Dr. Craig met Tycho Brahe and learned about the astronomers problems with multiplication Returned to Scotland and told his friend John Napier Returned to Scotland and told his friend John Napier Napier was inspired to invent logarithms – a tool that speeds calculation Napier was inspired to invent logarithms – a tool that speeds calculation

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Mirifici Logarithmorum Canonis Descriptio (1614) Written by John Napier and communicated logarithms to the world Written by John Napier and communicated logarithms to the world It took him 24 years to write It took him 24 years to write Napiers logarithms were quite different from modern logarithms but just as useful for computation Napiers logarithms were quite different from modern logarithms but just as useful for computation Napier, lord of Markinston, hath set upon my head and hands a work with his new and admirable logarithms. I hope to see him this summer, if it please God, for I never saw a book which pleased me better or made me more wonder. -- Henry Briggs (1561-1630)

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Logarithms are Exponents The two forms on the left are equivalent. The second is read y equals log base 2 of x.

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Logarithms are Exponents x Scientific Notationlog 10 x 0.00011 × 10 -4 -4 0.0011 × 10 -3 -3 0.011 × 10 -2 -2 0.11 × 10 -1 11 × 10 0 0 101 × 10 1 1 1001 × 10 2 2 10001 × 10 3 3 100001 × 10 4 4 A base 10 logarithm is written log 10 x A base 10 logarithm is written log 10 x For example: log 10 1000 = 3 For example: log 10 1000 = 3 The base 10 log expresses how many factors of ten a number is – its order of magnitude The base 10 log expresses how many factors of ten a number is – its order of magnitude

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Only positive numbers have logarithms log 10 0 = x is undefined because 10 x = 0 has no solution Notice that adding one to the base ten log is the same as multiplying the number by ten x Scientific Notation log 10 x (nearest thousandth ) 0.00015 4 1.54 × 10 -4 -3.8125 0.001541.54 × 10 -3 -2.8125 0.01541.54 × 10 -2 -1.8125 0.1541.54 × 10 -1 -0.8125 1.541.54 × 10 0 0.8125 15.41.54 × 10 1 1.8125 1541.54 × 10 2 2.8125 15401.54 × 10 3 3.8125 154001.54 × 10 4 4.8125

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Richter Magnitudes DescriptionEffects Frequency of Occurrence Less than 2.0 Micro Microearthquakes, not felt. About 8,000 per day 2.0-2.9Minor Generally not felt, but recorded. About 1,000 per day 3.0-3.9 Often felt, but rarely causes damage 49,000 per year (est.) 4.0-4.9Light Noticeable shaking of indoor items, rattling noises. Significant damage unlikely. 6,200 per year (est.) 5.0-5.9Moderate Can cause major damage to poorly constructed buildings over small regions. At most slight damage to well-designed buildings. 800 per year 6.0-6.9Strong Can be destructive in areas up to about 160 kilometers (100 mi) across in populated areas. 120 per year 7.0-7.9Major Can cause serious damage over larger areas. 18 per year 18 per year 8.0-8.9Great Can cause serious damage in areas several hundred miles across. 1 per year 9.0-9.9 Devastating in areas several thousand miles across. 1 per 20 years 10.0+Epic Never recorded Extremely rare (Unknown) The Richter Magnitude is an Exponent

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Kepler and Napier The time it takes for each planet to orbit the sun is related to its distance from the sun Kepler might not have seen this relationship if not for logarithmic scales as seen here This insight helped Newton discover his Law of Gravity

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Dimension We normally think of dimension as either 1D, 2D, or 3D

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How Long is a Coastline? The length of a coastline depends on how long your ruler is The ruler on the left measures a 6 unit coastline The rule on the right is half as long and measures a 7.5 unit coastline

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Fractal Dimension For any specific coastline, s is the length of the rule and L(s) is the length measured by the ruler. A log/log plot gives a straight line The equations on the right are for each line The fractal dimension of a coast is (1 - slope ) The more negative the slope, the rougher the coast Photo downloaded 5/12/10 from http://cruises.about.com/od/capetown/ig/Cape-Point/Cape-of-Good-Hope.htm

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Repeating Scales This is the Scottish coast All fractals are self similar – they have similar details at big scales and little scales Notice how the big bays are similar to the small bays, which are similar to the tiny inlets http://visitbritainnordic.wordpress.com/2009/06/09/british-history/

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The Koch Curve The Koch Curve has a fractal dimension of 1.26

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Cantor Dust Cantor Dust is created by removing the middle third of every line Cantor Dust has a fractal dimension of 0.63

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Sierpenski Carpet The Sierpenski Triangle is created by removing the middle third of each triangle The fractal dimension is 1.59

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Leonhard Euler (1708-1783) Did important work in: number theory, artillery, northern lights, sound, the tides, navigation, ship-building, astronomy, hydrodynamics, magnetism, light, telescope design, canal construction, and lotteries One of the most important mathematicians of all time Its said that he had such concentration that he would write his research papers with a child on each knee while the rest of his thirteen children raised uninhibited pandemonium all around him

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Leonhard Euler Introduced the modern notation for sin/cos/tan, the constant i, and used for summation Introduced the concept of a function and function notation y = f (x) Proved that 2 31 -1=2,147,483,647 is prime Solved the Basel problem by proving that

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The Number e e is a constant e 2.718145927 Euler was the first to use the letter e for this constant. Supposedly a through d were taken e appears in many parts of math

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e and Slope In calculus, youll learn how to find the slope of any function The slope of y=e x at any point (x, y) is simply y Its the only function with this property

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Eulers Formula For any real number x This leads to Eulers formula Called The Most Beautiful Mathematical Formula Ever

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How Many Primes? π(x) is the number of prime numbers less than x A good estimate for π(x) is xπ(x)π(x) estimate 1000168169 1000012291218 10000095929512 10000007849878030 10000000664579661459 10000000057614555740304

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References http://www.mathpages.com/rr/s8-01/8-01.htm http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/worksho p/Fractals.html http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/worksho p/Fractals.html http://primes.utm.edu/howmany.shtml http://en.wikipedia.org/wiki/Euler

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