Introducing Mathematics E-Book

Jerry Ravetz

• Herausgeber: Icon Books
• Kategorie: Wissenschaft und neue Technologien
• Serie: Graphic Guides
• Sprache: Englisch
• Veröffentlichungsjahr: 2015

Seitenzahl: 126

Beschreibung

What is mathematics, and why is it such a mystery to so many people? Mathematics is the greatest creation of human intelligence. It affects us all. We depend on it in our daily lives, and yet many of the tools of mathematics, such as geometry, algebra and trigonometry, are descended from ancient or non-Western civilizations.  Introducing Mathematics traces the story of mathematics from the ancient world to modern times, describing the great discoveries and providing an accessible introduction to such topics as number-systems, geometry and algebra, the calculus, the theory of the infinite, statistical reasoning and chaos theory. It shows how the history of mathematics has seen progress and paradox go hand in hand - and how this is still happening today.

Leseprobe

Published by Icon Books Ltd, Omnibus Business Centre, 39–41 North Road, London N7 9DP Email: [email protected]

ISBN: 978-1-84831-297-5

Text copyright © 2012 Icon Books Ltd

Illustrations copyright © 2012 Icon Books Ltd

The author and illustrator has asserted their moral rights

Originating editor: Richard Appignanesi

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

Contents

Cover

Title Page

Copyright

WHY MATHS?

COUNTING

WRITTEN NUMBERS

THE ZERO

SPECIAL NUMBERS

LARGE NUMBERS

POWERS

LOGARITHMS

CALCULATION

EQUATIONS

MEASUREMENT

GREEK MATHEMATICS

PYTHAGORAS

ZENO’S PARADOXES

EUCLID

CHINESE MATHEMATICS

THE CHIU CHANG

FOUR CHINESE MATHEMATICIANS

INDIAN MATHEMATICS

VEDIC GEOMETRY

BRAHMAGUPTA

JAIN NUMBERS

VEDIC AND JAIN COMBINATIONS

MATHEMATICAL VERSE

RAMANUJAN

ISLAMIC MATHEMATICS

AL-KHUWARAZMI

DEVELOPMENT OF ALGEBRA

THE DISCOVERY OF TRIGONOMETRY

AL-BATTANI

ABU WAFA

IBN YUNUS AND THABIT IBN QURRA

AL-TUSI

SOLUTIONS OF PROBLEMS INVOLVING INTEGERS

EMERGENCE OF EUROPEAN MATHEMATICS

RENÉ DESCARTES

ANALYTIC GEOMETRY

FUNCTIONS

THE CALCULUS

DIFFERENTIATION

INTEGRATION

BERKELY’S QUESTIONS

EULER’S GOD

NON-EUCLIDEAN GEOMETRIES

N-DIMENSION SPACES

EVARISTE GALOIS

GROUPS

BOOLEAN ALGEBRA

CANTOR AND SETS

CRISIS IN MATHEMATICS

RUSSELL AND MATHEMATICAL TRUTH

GÖDEL’S THEOREM

THE TURING MACHINE

FRACTALS

CHAOS THEORY

TOPOLOGY

NUMBER THEORY

STATISTICS

P-VALUES AND OUTLIERS

PROBABILITY

UNCERTAINTY

POLICY NUMBERS

MATHEMATICS AND EUROCENTRISM

ETHNOMATHEMATICS

MATHEMATICS AND GENDER

WHERE NOW?

FURTHER READING

The Authors

Index

WHY MATHS?

Everybody moans at the very mention of “maths”. People think that the world is divided into two kinds of folks. The “brainy” lot who understand mathematics but are not the kind of people one wants to meet at parties...

...and the rest of us! Just keep your eyes open for any mathematicians O.K.?

But all of us need to understand maths to some extent. Without mathematics, life would be inconceivable.

We need maths when we go shopping, check our bills, manage household finances... ...and run our businesses.

We need maths to build our houses...

...insure our cars, do our banking. We need maths to make maps so we can find our way around cities... ...travel around the world, even go out into space!

Thus, mathematics is the engine that runs our industrial civilization. It is the language of science, technology and engineering. It is essential for architecture and design as well as economics and medicine. Even art relies on mathematics to some extent.

Indeed, mathematics has become a guide to the world in which we live, the world which we shape and change, and of which we are a part. And as the world becomes more and more complex, and uncertainties in our environment become more urgent and threatening, we need mathematics to describe the risks we face and to plan our remedies.

The ability to deal with mathematics does require a special talent and skill – like any other field of human endeavour, such as dancing. Just as an accomplished ballet performance is sophisticated and exquisite, so is mathematics in its essence very elegant and beautiful.

But even though most of us cannot become fully-fledged ballet performers, all of us know what it is to dance and virtually all of us can dance. Similarly, all of us should know what mathematics is about, and be able to understand and handle certain basic steps.

Fear of maths is like of dancing. Both are overcome by a little bit of practice.

Music is the pleasure the human soul experiences from counting without being aware that it is counting.

COUNTING

To some extent, young beginners at mathematics retrace the steps of humanity in the development of mathematical knowledge.

At school, children learn to count, to calculate, and to measure. Once they have been learned, these techniques may seem “elementary”. But for the learners they are full of mystery.

The naming of numbers becomes an incantation, especially when we get to the bigger ones. Counting to a hundred becomes tedious, but getting to a thousand is like climbing a mountain! What is the last number, the biggest one of all?

If there isn’t such a thing, then what is there at the end?

How do we name the numbers, as we call them out one after another? Perhaps just a few numbers are enough. Some animals can recognize different collections up to five or seven – beyond that it’s just “many”. But if we know that numbers go on continuously, we can’t just keep inventing new names indefinitely as we go along.

The language of the Dakota Indians was not written down.

So we counted the years and marked special events in our history by keeping a winter count like this one.

It is made of cloth and the pictographs are drawn in black ink. Each year a new pictograph was added to show the main event of the past year.

The best way to systematize naming and counting is to have a “base”, a number that marks the beginning of counting again. The simplest base is just two. For example, the Gumulgal, an Australian indigenous people, counted like this:

This may seem primitive and tedious. But the base two, in the form of 0’s and 1’s ... ...is built into digital computers as the foundation of all their calculations.

The fingers of the hands are useful for defining bases. Some systems use five, more common is ten. But many other bases can be used. The old British currency had several: twelve (pence per shilling) and then twenty (shillings per pound) and even twenty-one (shillings per guinea!). Shop assistants needed to keep reckoning books by their sides. And when people bought in instalments, they might be told that their living-room suite cost 155 guineas, or 104 weekly payments of one pound, fifteen shillings and sevenpence-halfpenny.

Who could calculate the interest on that? Small wonder that instalment payments were called the “never-never” – –you never finish paying!

The base twenty (fingers and toes?) is also common. The Yoruba used this, employing subtraction for the larger numbers within the base. They had different names for the numbers one (okan) to ten (eewa). From eleven to fourteen, they simply added. So eleven became “one more than ten”, and fourteen “four more than ten”. But from fifteen onwards they subtracted. So fifteen became “twenty less five” and nineteen became “twenty less one”. The base twenty still survives in French, where eighty is “four-twenties”, and ninety-nine is “four-twenties-nineteen”.

Those who deal with computers use bases built on two.

So no single base is “best”. We can think of a number system as designed with different attributes: easy to remember, convenient in naming, useful for calculating, etc.

Once the grouping, or base, for a number system had been developed, it allowed the four basic functions of arithmetic... ...to be developed easily.

WRITTEN NUMBERS

It is possible to count effectively in a culture with no writing. But calculating then requires much memory and special skills. As writing spread among civilizations, different systems, some quite sophisticated, emerged.

The Aztecs used a system based on 20, with four basic symbols.

1 was represented with a blob designating a maize-seed pod.

20 was represented with a flag.

400 was designated by a maize plant.

8000 was symbolized by a maize dolly.

These symbols could be used to represent all kinds of numbers. For example, the number 9287 was represented as:

The Mayans’ numbering system had only three symbols:

...a large dot • was one, ...a bar____was fine, ...and a snail’s shell was zero.

The Ancient Egyptians (c. 4000–3000 BC) used a pictorial script (hieroglyph) to write down their numbers.

The pictograms, starting with one, increased by ten times, eventually reaching ten million.

The Babylonians (c. 2000 BC) used a system based on 60 and its multiples, with the following symbols:

Later, they evolved a system based on only two values:

So, 95 would be written as

You know I’ve lost count of the number of my wives... Yes... As a Babylonian, I was able to spend nearly a whole extra hour in bed this morning... I’ll go to the foot of our stairs!

The Babylonian sexagesimal system has survived to this day. Circles have 360 degrees. Hours have sixty minutes. Minutes have sixty seconds.

The Ancient Chinese (c. 1400–1100 BC) used a base 10 system of numbers with symbols for one to ten, a hundred, a thousand and ten thousand. Later, around the 3rd century BC, the Chinese developed a form of numerals using straight lines (or rods),

So! It’s a typical Oriental stereotype.

The Chinese made the great invention that put written symbols in a different world from the spoken names of numbers. This was a system of “place-value”. The meaning of a number, as an expression of quantity, depended on its place in the string of numbers. Thus “2” could mean two, twenty, or two hundred, depending on its location. This made it unnecessary to name the higher bases – in “234” we know that the 2 means 200.

Elementary, my dear Watson. The number 2.689 is shown here with each figure proportionately-sized to show the quantity it represents! Hence the “value” of “place”... The figure “9” is so small I’m needed to make it readable!

The Indians developed three distinct types of number systems.

The Kharosthi (c. 400–200 BC) used symbols for ten and twenty, and numbers up to one hundred were built up by addition.

The Brahmi (c. 300 BC) used separate symbols for the digits one, four, to nine and ten, a hundred, a thousand and so on.

The Gwalior (c. 850 AD) had symbols for numbers one to nine as well as for zero.

Think of a number... Ok., now double it... treble it... quadrupeddle it...

The Indians were very comfortable with high numbers. The classical Hindu texts give names to numbers as large as 1,000,000,000,000 (parardha)!

The Ancient Greeks (c. 900 BC-200 AD) had two parallel systems. The first was based on the initial letters of the names of the 17 numbers. So, five was symbolized by the letter pi, ten by the letter delta, one hundred by the antique form of letter H, and so on. The second system, which emerged around the 3rd century BC, used all the letters of the Greek alphabets and three from the Phoenician alphabet, making a total of twenty-seven numerical symbols. The first nine letters of the alphabet signified the numbers 1 to 9; the second nine letters were used for tens from 10 to 90; and the last nine letters described the hundreds from 100 to 900.

We Greeks fought shy of large numbers, and our terminology hardly took us beyond the “myriad” (10.000).

The Roman system (400 BC-600 AD) had a total of seven symbols: I for 1, V for 5, X for 10, L for 50, C for 100, D for 500 and M for 1000.

The numbers are written from left to right with the largest quantities placed at the left and added together to obtain the designated number.

So, LX is 60.

For convenience, a smaller quantity on the left was interpreted as a subtraction. So, MCM means 1900.

The Roman numerals, still used today for ornament, were not suitable for doing rapid calculations.

Oi! Clock-face!

The use of the alphabet for numbers enabled the rise of a highly developed art of divination called “gematria”. Given any word, or particularly a name, one would rearrange the letters to form a number and then scrutinize that for its quality and meaning. Anyone whose name yielded 666 (the Biblical “number of the beast”) was obviously a Bad Thing!

Only with me, Descartes, and my successors in Europe did mathematics become completely “disenchanted”, at least for the educated elite. Now stop all this nonsense, or I’ll put a curse on you.

Bad news, good young sirrah! Your name has the number “Spawn Of Satan”. I have the remedy here in my hand, my lady.

It was said that in World War II, resistance against me among some fundamentalist Christians was strengthened by the discovery that I was a 666 type. Are you looking at my pint?

The Muslim civilization (650 AD-present) developed two sets of numerals. The sets were similar, but one was used in the eastern part of the Muslim world (Arabia and Persia), while the other was common in the western part (the Maghrib and Muslim Spain). Both contained ten symbols from zero to nine.

Eastern set:

Western set: 1 2 3 4 5 6 7 8 9 0

The eastern set is still used throughout the Arab world. The western set is what we now know as “Arabic numerals” – the system we all use today.

The next bus is a 971, I think. But surely, it’s a 250, is it not? Don’t ask me, I don’t do all this new-fangled Western stuff.

THE ZERO

The Zero is a relatively late invention (about 6th century AD), and it seems to have been a joint product of the Chinese and Hindu civilizations. The Chinese needed some such thing for their place-value notation – how would they represent the missing place for the number “two hundred and five”? Just 25 is wrong, so they had something to “fill” the empty place, as 2–5. But the full meaning of zero was developed in the Indian civilization, where philosophical speculation on The Void was highly developed.

The answer’s a big nothing.

That sort of cultural background was very necessary for the invention, for the Zero is very peculiar. In some ways it behaves like other numbers, for we can add with it.

While zero is essential for calculation, it is excluded from counting. The first in a row of things is not the “zero-th”. This paradox shows up in the calendar: the 1900s are the 20th century, since there was no zero-th century at the start of the Western calendar A.D.

Also, zero has two meanings, as we see from “the fossils joke”. A museum guide talks to a school party:

This fossil bone is sixty-five million and four years old. How do you know so exactly? Well, when I came to this job I was told that it was 65,000,000 years old... and that was four years ago.

Of course everyone saw that this was ridiculous, but one of the pupils did the sum...

SPECIAL NUMBERS