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Why Logs?

Why Logs?. From Calculating to Calculus. John Napier (1550-1617). Scottish mathematician, physicist, astronomer/astrologer 8th Laird (baron) of Merchistoun Famous for inventing logarithms

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Why Logs?

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  1. Why Logs? From Calculating to Calculus

  2. John Napier(1550-1617) • Scottish mathematician, physicist, astronomer/astrologer • 8th Laird (baron) of Merchistoun • Famous for inventing logarithms • Before digital computers, logarithms were vital for computation, at a time when “computers” were people • 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

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

  4. Island of HvenTycho Brahe’s Playground • Built for Brahe by the King of Denmark at great expense • Active observatory from 1576-1580 • Hosted wild and crazy parties • The island had its own zoo

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

  6. Mirifici Logarithmorum Canonis Descriptio (1614) • Written by John Napier and communicated logarithms to the world • It took him 24 years to write • Napier’s 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)

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

  8. Logarithms are Exponents • A base 10 logarithm is written log10x • For example: log10 1000 = 3 • The base 10 log expresses how many factors of ten a number is – its “order of magnitude”

  9. Only positive numbers have logarithms • log10 0 = x is undefined because 10x = 0 has no solution • Notice that adding one to the base ten log is the same as multiplying the number by ten

  10. The Richter Magnitude is an Exponent

  11. 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

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

  13. 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

  14. 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

  15. 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 Repeating Scales http://visitbritainnordic.wordpress.com/2009/06/09/british-history/

  16. The Koch Curve • The Koch Curve has a fractal dimension of 1.26

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

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

  19. 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 • It’s 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

  20. 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 231-1=2,147,483,647 is prime • Solved the Basel problem by proving that

  21. The Number e • eis a constant • e≈ 2.718145927 • Euler was the first to use the letter efor this constant. Supposedly athrough dwere taken • e appears in many parts of math

  22. e and Slope • In calculus, you’ll learn how to find the slope of any function • The slope of y=exat any point (x, y) is simply y • It’s the only function with this property

  23. Euler’s Formula • For any real number x • This leads to Euler’s formula • Called “The Most Beautiful Mathematical Formula Ever”

  24. How Many Primes? • π(x) is the number of prime numbers less than x • A good estimate for π(x) is

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

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