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