The Gentle Art of Mathematics

by Dan Pedoe

Overview

This lighthearted work uses a variety of practical applications and puzzles to take a look at today's mathematical trends. In nine chapters, Professor Pedoe covers mathematical games, chance and choice, automatic thinking, and more.

Product Details

ISBN-13:	9780486164069
Publisher:	Dover Publications
Publication date:	11/29/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	160
File size:	5 MB

Table of Contents

CHAPTER I MATHEMATICAL GAMES
The fascination of ordinary numbers
Why fifteen Fellows of the Royal Society?
The scale of ten
A problem involving ordinary numbers
A very long division
A much shorter solution of the digital problem
"Sixteen months in the year, and their names"
"The binary scale, or scale of two"
A magic table of numbers
The game of Nim
As played by an electronic brain against humans
The theory behind the game
Winning positions in the game
Punched cards and automatic rearrangement of twelve cards
The twelve-coin problem
Can it be done without the use of mathematics?
"The ternary scale, or scale of three"
A solution of the twelve-coin problem
Weighing up to forty pounds with only four weights
There is an infinity of prime numbers
The square root of two is not a rational number
CHAPTER II CHANCE AND CHOICE
A coin is spun
Dr. Joad and the law of averages
Historical background to theory of probability
What is random behaviour?
Scattering seed at random
Urns and dice
Addition law of probabilities
Multiplication law
Errors of mathematicians
Eliza Doolittle
Odds that a head turns up in tossing a penny
A problem of Samuel Pepys
Two letters from Isaac Newton to Samuel Pepys
Expectation of a prize in a football pool
Expectation of eternal bliss
The St. Petersburg problem
Moral criticism of mathematical results
"Buffon's test, using child labour"
The courageous Bertrand
Anything which can happen will happen
Buffon's needle theorem and the evaluation of p
The giddy Lazzerini
Extra-sensory perception and psycho-kinesis
Why does heads turn up when you pray for tails?
CHAPTER III WHERE DOES IT END?
Is infinity greater than infinity?
Can you count?
Definition of an infinite class
Countable infinities
The positive rationals can be counted
The decimals greater than zero and less than one cannot be counted
A great unsolved problem of mathematics
The terrible Cantor
CHAPTER IV AUTOMATIC THINKING
Classes
One class contained in another
Syllogisms
Socrates was mortal
Universal class and null class
Some laws are unsatisfactory
Writers and Shakespeare
Another Lewis Carroll teaser
Algebra of classes and propositions
"Alice, Brenda, Cissie and Doreen"
Who won the scholarship?
CHAPTER V TWO-WAY STRETCH
Ballon d'essai
Rubber-sheet geometry
Topological transformation defined
Deformations
The escape-artist's trick
Supplying three houses with main services
Is topology worth while?
Multiply-connected figures
Sphere and torus
The Moebius band
"Fun with paper, gum and scissors"
Rotating ring of tetrahedra
Modern art and the Klein bottle
Simple polyhedra and Euler's formula
The four-colour theorem
Can you prove it?
Disdainful doggerel
CHAPTER VI RULES OF PLAY
Laws of addition
A double negative gives a positive
Additive groups
"What every airman knows, or how to add vectors"
Rotation is addition
Finite groups
How to multiply
Rings (not of commercial firms)
The Pascal triangle
The binomial theorem
"Perms. and combs., or how to arrange and select"
No help with football-pools
How to divide
Why exclude division by zero?
The group postulates
Do you put your shirt on before your tie?
A plane slides over itself
Symmetry investigated
Inkblots rationalised
Rotational symmetry
Ornaments
Point-lattices and curtain materials
The symmetries in Arabic art
CHAPTER VII AN ACCOUNTANT'S NIGHTMARE
The gullible Emperor
A fable of a slowly but surely divergent series
A well-behaved series
Can you rub out this line?
Decimals which come to an end
Those which do not
What kind of decimals arise from rational numbers
The uniqueness of infinite decimals
Irrational numbers
The number p
Shanks and p
A mystic rhyme for p
Why should seven suffer?
"Sir your superior mathematics"
Trouble with series
Pinning them down
More fuss and bother
Safety first
Achilles and the tortoise
Is he still running?
CHAPTER VIII DOUBLE TALK
Mathematicians not logical
The uncertainty of logic
Paradoxes galore
Class of all classes paradox
A humble mathematician
Mathematics not logic
Infinite collections of shoes
Of socks
Can you choose?
Intuitionism
Law of the excluded middle
Right or wrong?
Formalist view
No neurosis amongst mathematicians
CHAPTER IX WHAT IS MATHEMATICS?
What mathematicians do
International conferences
Mathematicians as human beings
What mathematics is not
Poincaré to the rescue