Why Logs?Why Logs?
From Calculating to CalculusFrom Calculating to Calculus
John NapierJohn Napier(1550-1617)(1550-1617)
Scottish mathematician, Scottish mathematician, physicist, physicist, astronomer/astrologerastronomer/astrologer
8th Laird (baron) of 8th Laird (baron) of MerchistounMerchistoun
Famous for inventing Famous for inventing logarithmslogarithms
Before digital computers, Before digital computers, logarithms were vital for logarithms were vital for computation, at a time when computation, at a time when “computers” were “computers” were peoplepeople
Slide rulesSlide rules are hand computers are hand computers based on logarithmsbased on logarithms
Slide rule image downloaded 5-11-10 from http://en.wikipedia.org/wiki/File:Pocket_slide_rule.jpg
Tycho BraheTycho Brahe(1546-1601)(1546-1601)
Born at Knutstorp Castle in Born at Knutstorp Castle in DenmarkDenmark
Meticulous observer of the Meticulous observer of the stars and planetsstars and planets
Led the way to proving that Led the way to proving that the earth revolves around the the earth revolves around the sunsun
Lived on the Island of HvenLived on the Island of Hven Lost part of his nose in a duelLost part of his nose in a duel
Island of HvenIsland of HvenTycho Brahe’s PlaygroundTycho Brahe’s Playground
Built for Brahe by the Built for Brahe by the King of Denmark at King of Denmark at great expensegreat expense
Active observatory Active observatory from 1576-1580from 1576-1580
Hosted wild and crazy Hosted wild and crazy partiesparties
The island had its own The island had its own zoozoo
Dr. John CraigDr. John Craig(? – 1620)(? – 1620)
In 1590 Dr. Craig was travelling In 1590 Dr. Craig was travelling with James VI of Scotland when with James VI of Scotland when he was shipwrecked at Hvenhe was shipwrecked at Hven
The incident may have inspired The incident may have inspired Shakespeare’s Shakespeare’s The TempestThe Tempest
Dr. Craig met Tycho Brahe and Dr. Craig met Tycho Brahe and learned about the astronomer’s learned about the astronomer’s problems with multiplicationproblems with multiplication
Returned to Scotland and told his Returned to Scotland and told his friend John Napierfriend John Napier
Napier was inspired to invent Napier was inspired to invent logarithms – a tool that speeds logarithms – a tool that speeds calculationcalculation
Mirifici Logarithmorum Canonis Mirifici Logarithmorum Canonis DescriptioDescriptio (1614) (1614)
Written by John Napier and Written by John Napier and communicated logarithms to the worldcommunicated logarithms to the worldIt took him 24 years to writeIt took him 24 years to writeNapier’s logarithms were quite Napier’s logarithms were quite different from modern logarithms but different from modern logarithms but just as useful for computationjust as useful for computationNapier, lord of Markinston, hath set upon my head and hands Napier, lord of Markinston, hath set upon my head and hands a work with his new and admirable logarithms. I hope to see 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 him this summer, if it please God, for I never saw a book which pleased me better or made me more wonder. -- which pleased me better or made me more wonder. -- Henry Briggs (1561-1630)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
xScientificNotation log10 x
0.0001 1 × 10-4 -4
0.001 1 × 10-3 -3
0.01 1 × 10-2 -2
0.1 1 × 10-1 -1
1 1 × 100 0
10 1 × 101 1
100 1 × 102 2
1000 1 × 103 3
10000 1 × 104 4
A base 10 logarithm is A base 10 logarithm is written logwritten log1010 xx
For example:For example:loglog1010 1000 = 3 1000 = 3
The base 10 log The base 10 log expresses how many expresses how many factors of ten a number factors of ten a number is – its “order of is – its “order of magnitude”magnitude”
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
xScientificNotation
log10 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 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
Richter Richter MagnitudesMagnitudes
DescriptionDescription EffectsEffects Frequency of Frequency of OccurrenceOccurrence
Less than 2.0Less than 2.0 MicroMicro Microearthquakes, not felt.Microearthquakes, not felt. About 8,000 per dayAbout 8,000 per day
2.0-2.92.0-2.9 MinorMinor Generally not felt, but recorded.Generally not felt, but recorded. About 1,000 per day
3.0-3.93.0-3.9 Often felt, but rarely causes damageOften felt, but rarely causes damage 49,000 per year (est.)
4.0-4.94.0-4.9 LightLight Noticeable shaking of indoor items, rattling Noticeable shaking of indoor items, rattling noises. Significant damage unlikely.noises. Significant damage unlikely.
6,200 per year (est.)
5.0-5.95.0-5.9 ModerateModerate Can cause major damage to poorly constructed Can cause major damage to poorly constructed buildings over small regions. At most slight buildings over small regions. At most slight damage to well-designed buildings.damage to well-designed buildings.
800 per year
6.0-6.96.0-6.9 StrongStrong Can be destructive in areas up to about 160 Can be destructive in areas up to about 160 kilometers (100 mi) across in populated areas.kilometers (100 mi) across in populated areas.
120 per year
7.0-7.97.0-7.9 MajorMajor Can cause serious damage over larger areas. Can cause serious damage over larger areas. 18 18 per yearper year
18 per year
8.0-8.98.0-8.9 GreatGreat Can cause serious damage in areas several Can cause serious damage in areas several hundred miles across.hundred miles across.
1 per year
9.0-9.99.0-9.9 Devastating in areas several thousand miles Devastating in areas several thousand miles across.across.
1 per 20 years
10.0+10.0+ EpicEpic Never recordedNever recorded Extremely rare (Unknown)
The Richter Magnitude is an Exponent
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
Dimension
We normally think of dimension as either 1D, 2D, or 3D
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
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
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/
The Koch Curve
The Koch Curve has a fractal dimension of 1.26
Cantor Dust
Cantor Dust is created by removing the middle third of every line
Cantor Dust has a fractal dimension of 0.63
Sierpenski Carpet
The Sierpenski Triangle is created by removing the middle third of each triangle
The fractal dimension is 1.59
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
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
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
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
For any real number x
This leads to Euler’s formula
Called “The Most Beautiful Mathematical Formula Ever”
How Many Primes?
π(x) is the number of prime numbers less than x
A good estimate for π(x) is
1log x
x
x π(x) estimate
1000 168 169
10000 1229 1218
100000 9592 9512
1000000 78498 78030
10000000 664579 661459
100000000 5761455 5740304
1ln x
x
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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