30-Second Maths E-Book

Richard J. Brown

• Herausgeber: Icon Books
• Kategorie: Wissenschaft und neue Technologien
• Serie: 30-Second
• Sprache: Englisch

Seitenzahl: 152

Veröffentlichungsjahr: 2012

Beschreibung

From Rubik's cubes to Godel's incompleteness theorem, everything mathematical explained, with colour illustrations, in half a minute. Maths is enjoying a resurgence in popularity. So how can you avoid being the only dinner guest who has no idea who Fermat was, or what he proved? The more you know about Maths, the less of a science it becomes. 30 Second Maths takes the top 50 most engaging mathematical theories, and explains them to the general reader in half a minute, using nothing more than two pages, 200 words and one picture. Read at your own pace, and discover that maths can be more fascinating than you ever imagined.

Leseprobe

30-SECOND

MATHS

The 50 most mind-expanding theories in mathematics, each explained in half a minute

Editor

Richard Brown

Contributors

Richard Brown

Richard Elwes

Robert Fathauer

John Haigh

David Perry

Jamie Pommersheim

First published in the UK in 2012 by Icon Books Ltd

Omnibus Business Centre

39–41 North Road, London N7 9DP

email: [email protected]

www.iconbooks.co.uk

© 2012 by Ivy Press Limited

The editor and contributors have asserted their moral rights.

No part of this book may be reproduced in any form, or by any means, without prior permission in writing from the publisher.

This book was conceived, designed and produced by

Ivy Press

210 High Street, Lewes

East Sussex, BN7 2NS, UK

www.ivypress.co.uk

Creative Director Peter Bridgewater

Publisher Jason Hook

Editorial Director Caroline Earle

Art Director Michael Whitehead

Designer Ginny Zeal

Illustrator Ivan Hissey

Profiles Text Viv Croot

Glossaries Text Steve Luck

Project Editor Jamie Pumfrey

Digital Assistant Emily Owen

ISBN: 978-1-84831-386-6

Colour origination by Ivy Press Reprographics

CONTENTS

Introduction

Numbers & Counting

GLOSSARY

Fractions & Decimals

Rational & Irrational Numbers

Imaginary Numbers

Counting Bases

Prime Numbers

Fibonacci Numbers

Pascal’s Triangle

profile: Blaise Pascal

Number Theory

Making Numbers Work

GLOSSARY

Zero

Infinity

Addition & Subtraction

Multiplication & Division

Exponentials & Logarithms

Functions

profile: Gottfried Leibniz

Calculus

Chance is a Fine Thing

GLOSSARY

Game Theory

Calculating the Odds

profile: Girolamo Cardano

The Law of Large Numbers

The Gambler’s Fallacy – Law of Averages

The Gambler’s Fallacy – Doubling Up

Randomness

Bayes’ Theorem

Algebra & Abstraction

GLOSSARY

The Variable Placeholder

The Equation

Polynomial Equations

profile: Abu ‘Abdallah Muhammad ibn Musa al-Khwarizmi

Algorithms

Sets & Groups

Rings & Fields

Geometry & Shapes

GLOSSARY

Euclid’s Elements

Pi – The Circle Constant

The Golden Ratio

profile: Pythagoras

Trigonometry

Squaring the Circle

Parallel Lines

Graphs

Another Dimension

GLOSSARY

Platonic Solids

Topology

Euler Bricks

The Möbius Strip

profile: Archimedes of Syracuse

Fractals

Origami Geometry

Rubik’s Cube

Knot Theory

Proofs & Theorems

GLOSSARY

Fermat’s Last Theorem

profile: Pierre de Fermat

The Four Colour Mapping Problem

Hilbert’s Program

Gödel’s Incompleteness Theorem

Poincaré’s Conjecture

The Continuum Hypothesis

Riemann’s Hypothesis

APPENDICES

Resources

Notes on Contributors

Index

Acknowledgements

INTRODUCTION

Richard Brown

Elegant geometry

It is said that mathematics is the art of pure reason. It is the fundamental logical structure of all that exists, and all that doesn’t exist, in this reality of ours. Far removed from the simple calculations that allow us to balance our accounts and calculate our everyday affairs, mathematics helps us to order and understand the very notion of everything we can imagine in life. Like music, art and language, the essential symbols and concepts of mathematics, many of which are defined and discussed in this book, allow us to express ourselves in amazingly intricate ways and to define unimaginably complex and beautiful structures. While the practical uses for mathematics are rife, what makes mathematics so magical is its elegance and beauty outside of any real application. We give the concepts in mathematics meaning only because they make sense and help us to order our existence. But outside of the meaning we give these elements of maths, they do not really exist at all except in our imagination.

The natural and social sciences use mathematics to describe their theories and provide structure to their models, and arithmetic and algebra allow us to conduct our business and learn how to think. But beyond these practical applications lies the true nature of the discipline; mathematics is the framework and provides the rules of engagement for the entire system of structured thought.

This text is a glimpse into the world a mathematician sees in everyday life. Herein lies a set of some of the more basic and fundamental elements in the field today, with definitions, a little history, and some insight into the nature of many basic mathematical concepts. This book contains 50 entries, each of which centres on an important topic in mathematics. They are ordered into seven categories, which roughly help to define their context. In Numbers & Counting, we explore the basic building blocks that allow us to enumerate our surroundings. We study some of the operations and structures on numbers in Making Numbers Work. These entries basically describe the arithmetic system that helps us to use mathematics in our everyday lives. In Chance is a Fine Thing, we detail some ideas and consequences when using mathematics to understand random events and chance happenings. Next, we lay out some of the deeper, more complex structures of numbers in Algebra & Abstraction. It is here that the path towards higher mathematics begins. In turn, we explore the more visual aspects of mathematical relationships in Geometry & Shapes. Since mathematical abstraction is one of pure imagination, we then explore what happens outside of our three dimensions in Another Dimension. And finally, in Proofs & Theorems, we discuss some of the more profound ideas and facts that our mathematical path has led us to.

Individually, each entry is a brief glimpse into one of the more beautiful and important ideas central to mathematics today. Each topic is presented in the same format, aimed at facilitating a proper introduction; the 3-second sum offers the briefest overview, the 30-second maths goes into further depth on the topic, and a 3-minute addition begins the process of pondering the deeper connections between the idea and its importance in the world. It is hoped that, taken together, these elements will help to open your eyes to a closer understanding of the nuts and bolts of what mathematics is really all about.

When used as a reference text, this book will provide the basics of some of the more profound ideas in mathematics. When read in full, this text may provide a glimpse into another world as rich and meaningful as the one you live in now: the world of mathematics.

Dimensional beauty

There are only five ways to construct a three-dimensional solid using regular polygons. It is not hard to see why. But does that make these objects special? Mathematicians think so.

NUMBERS & COUNTING

NUMBERS & COUNTING

GLOSSARY

algebra One of the main branches of pure mathematics which studies operations and relations on numbers. Elementary algebra involves studying the rules of arithmetic on expressions involving variables. Advanced algebra involves studying these operations and relations on mathematical objects and constructions other than numbers.

complex number Any number that comprises both real and imaginary number components, such as a + bi, in which a and b represent any real number and i represents √−1. See imaginary number.

factor One of two or more numbers that divides a third number exactly. For example 3 and 4 are factors of 12, as are 1, 2, 6 and 12.

figurate number Any number that can be represented as a regular geometric shape, such as a triangle, square or hexagon.

fractional number (fraction) Any number that represents part of a whole. The most common fractions are called common or vulgar fractions, in which the bottom number, the denominator, is a non-zero integer denoting how many parts make up the whole, whereas the top number, the numerator, represents the number of equal divisions of the whole. Proper fractions represent a value of less than 1, e.g., ⅔, whereas improper fractions represent a value greater than 1, e.g., 3/2, or 1⅓.

integer Any natural number (the counting numbers 1, 2, 3, 4, 5 and so on), 0 or the negative natural numbers.

irrational number Any number that cannot be expressed as a ratio of the integers on a number line. The most commonly cited examples of irrational numbers are π and √2. A good way of identifying an irrational number is to check that its decimal expansion does not repeat. Most real numbers are irrational numbers.

number line The visual representation of all real numbers on a horizontal scale, with negative values running indefinitely to the left and positive to the right, divided by zero. Most number lines usually show the positive and negative integers spaced evenly apart.

polynomial An expression using numbers and variables, which only allows the operations of addition, multiplication and positive integer exponents, i.e., x2. (See Polynomial Equations.)

rational number Any number that can be expressed as a ratio of the integers on a number line; or more simply any number that can be written as a fraction, including whole numbers. Rational numbers are also identified by finite or repeating decimals.

real number Any number that expresses a quantity along a number line or continuum. Real numbers include all of the rational and the irrational numbers.

whole number Also known as a natural or counting number, a whole number is any positive integer on a number line or continuum. Opinion varies, however, on whether 0 is a whole number.

FRACTIONS & DECIMALS

the 30-second maths

The whole numbers, 0, 1, 2, 3…, are the bedrock of mathematics, and have been used by humans for millennia. But not everything can be measured using whole numbers. If 15 hectares of land are divided between 7 farmers, each will have 15/7 (or 21/7) hectares. The simplest non-whole numbers can be expressed in a fractional form like this. But for other numbers, such as π, this is awkward or impossible. With the development of science came the need to subdivide quantities ever more accurately. Enter the decimal system, an efficient column-based method using Hindu-Arabic numerals. Here, the number 725 has three columns, and stands for 7 hundreds, 2 tens and 5 units. By adding a decimal point after the units, and extra columns to its right, this approach easily extends to numbers smaller than a unit. So 725.43 stands for 7 hundreds, 2 tens, 5 units, 4 tenths (of a unit) and 3 hundredths. By incorporating ever more columns to the left or to the right, numbers both large and small can be written as precisely as needed. In fact every number in between the whole numbers can be expressed as a decimal (but not as a fraction), giving us the ‘real’ number system.

3-SECOND SUM

The starting point for mathematics is the system of whole numbers, 0, 1, 2, 3… But many things fall between the gaps, and there are two ways to measure them.

3-MINUTE ADDITION

Translating between fractions and decimals is not always straightforward. It is easy to recognize 0.25, 0.5 and 0.75 as ¼, ½ and ¾ respectively. But the decimal equivalent of ⅓, is 0.333333…, where the string of 3s never ends, and 1/7 is 0.142857142857142857…, also with a never-ending repeating pattern. It turns out all fractional numbers have repeating patterns in their decimal, while non-fractional numbers like π have decimals that do not repeat. These are the irrational real numbers.

RELATED THEORIES

RATIONAL & IRRATIONAL NUMBERS

COUNTING BASES

ZERO

3-SECOND BIOGRAPHIES

ABU 'ABDALLAH MUHAMMAD IBN MUSA AL-KHWARIZMI

c. 790–850

ABU’L HASAN AHMAD IBN IBRAHIM AL-UQLIDISI

c. 920–980

IBN YAHYA AL-MAGHRIBI AL-SAMAWAL

c. 1130–1180

LEONARDO PISANO (FIBONACCI)

c. 1170–1250

30-SECOND TEXT

Richard Elwes

Whole numbers can be subdivided into fractions, and decimals express these divisions even more precisely.

RATIONAL & IRRATIONAL NUMBERS

the 30-second maths

3-SECOND SUM

‘Real’ numbers — the numbers used to express quantities and representable via a decimal expansion — are either rational or irrational. But some irrationals are more unusual than others.

3-MINUTE ADDITION

The philosophy of the ancient Greeks held that all things measurable are, at worst, the ratio of whole numbers. Anecdotal history holds that the Pythagoreans were so distraught to discover that √2 is irrational that Hippasus of Metapontum was murdered to prevent revelation of this truth to the world. A number like π is perhaps more intuitively irrational, but it was only 250 years ago that this was proved true, and another century would pass before π was proved to be transcendental.

RELATED THEORIES

FRACTIONS & DECIMALS

EXPONENTIALS & LOGARITHMS

POLYNOMIAL EQUATIONS

PI – THE CIRCLE CONSTANT

PYTHAGORAS

3-SECOND BIOGRAPHIES

HIPPASUS OF METAPONTUM

active fifth century BCE

JOHANN LAMBERT

1728–1777

CHARLES HERMITE

1822–1901

FERDINAND VON LINDERMANN

1852–1939

30-SECOND TEXT

David Perry

Be real – numbers are rational if they can be written as a fraction. Otherwise they are irrational.

IMAGINARY NUMBERS

the 30-second maths

3-SECOND SUM

Today’s mathematicians work in an expanded number system, which includes a new ‘imaginary’ number i, the square root of −1.

3-MINUTE ADDITION

RELATED THEORIES

FRACTIONS & DECIMALS

POLYNOMIAL EQUATIONS

RIEMANN’S HYPOTHESIS

3-SECOND BIOGRAPHIES

NICCOLÒ FONTANA (‘TARTAGLIA’)

1500–1557

GIROLAMO CARDANO

1501–1576

RAFAEL BOMBELLI

1526–1572

CARL-FRIEDRICH GAUSS

1777–1855

AUGUSTIN-LOUIS CAUCHY

1789–1857

30-SECOND TEXT

Richard Elwes

Positive and negative integers weren’t enough for some mathematicians – they needed imaginary numbers.

COUNTING BASES

the 30-second maths

When we count numbers beyond