Why Logs From Calculating to Calculus John Napier

Why Logs? From Calculating to Calculus

John Napier (1550 -1617) n n n Scottish mathematician, physicist, astronomer/astrologer 8 th 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 511 -10 from http: //en. wikipedia. org/wiki/File : Pocket_slide_rule. jpg

Tycho Brahe (1546 -1601) n n n 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

Island of Hven Tycho Brahe’s Playground n n 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

Dr. John Craig (? – 1620) n n n 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 told his friend John Napier was inspired to invent logarithms – a tool that speeds calculation

Mirifici Logarithmorum Canonis Descriptio (1614) Written by John Napier and communicated logarithms to the world n. It took him 24 years to write n. Napier’s logarithms were quite different from modern logarithms but just as useful for computation n 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)

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

Logarithms are Exponents n n n A base 10 logarithm is written log 10 x For example: log 10 1000 = 3 The base 10 log expresses how many factors of ten a number is – its “order of magnitude” x Scientific Notation log 10 x 0. 0001 0. 001 1 × 10 -4 1 × 10 -3 -4 -3 0. 01 0. 1 1 10 10000 1 × 10 -2 1 × 10 -1 1 × 100 1 × 101 1 × 102 1 × 103 1 × 104 -2 -1 0 1 2 3 4

Only positive numbers have logarithms n n 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 Scientific Notation x log 10 x (nearest thousandth) 0. 000154 1. 54 × 10 -4 -3. 8125 0. 00154 1. 54 × 10 -3 -2. 8125 0. 0154 1. 54 × 10 -2 -1. 8125 0. 154 1. 54 × 10 -1 -0. 8125 1. 54 × 100 0. 8125 15. 4 1. 54 × 101 1. 8125 154 1. 54 × 102 2. 8125 1540 1. 54 × 103 3. 8125 15400 1. 54 × 104 4. 8125

The Richter Magnitude is an Exponent Richter Magnitudes Description Effects Frequency of Occurrence Less than 2. 0 Microearthquakes, not felt. About 8, 000 per day 2. 0 -2. 9 Minor Generally not felt, but recorded. About 1, 000 per day 3. 0 -3. 9 Often felt, but rarely causes damage 4. 0 -4. 9 Light Noticeable shaking of indoor items, rattling noises. 6, 200 per year (est. ) Significant damage unlikely. 5. 0 -5. 9 Moderate 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. 9 Strong Can be destructive in areas up to about 160 kilometers (100 mi) across in populated areas. 120 per year 7. 0 -7. 9 Major Can cause serious damage over larger areas. per year 8. 0 -8. 9 Great Can cause serious damage in areas several hundred miles across. 1 per year Devastating in areas several thousand miles across. 1 per 20 years Never recorded Extremely rare (Unknown) 9. 0 -9. 9 10. 0+ Epic 49, 000 per year (est. ) 18 18 per year

Kepler and Napier n n n 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

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

How Long is a Coastline? n n n 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

Fractal Dimension n n 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

Repeating Scales n n n 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/

The Koch Curve n The Koch Curve has a fractal dimension of 1. 26

Cantor Dust n n Cantor Dust is created by removing the middle third of every line Cantor Dust has a fractal dimension of 0. 63

Sierpenski Carpet n n The Sierpenski Triangle is created by removing the middle third of each triangle The fractal dimension is 1. 59

Leonhard Euler (1708 -1783) n 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 n n 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

Leonhard Euler n n 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

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

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

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

How Many Primes? n n π(x) is the number of prime numbers less than x A good estimate for π(x) is x estimate π(x) 1000 168 169 10000 1229 1218 100000 9592 9512 1000000 78498 78030 10000000 664579 661459 10000 5761455 5740304

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