The Story of Mathematics

STORY OF

EMATICS

.......... STORY OF

EMATICS

From creating the pyramids to exploring infinity

Anne Rooney

fIl

ARCTURUS

Aclmowledgements Wim thllnk, rv th= of my Flluhook f.und!;rho hllre htlped in i'llriOU! "".ryr 7J

A

A

~

)l ?

~

!

"i

-

.::.I-

,

,

t.

=?

rt

51

llll

-

~

? ?," /3~.3 ~

1 ?

~

!!l; ~

~

~

Egyptlllll burnt/(' mflflt'rflir qfrbe New Killgdllm (l600-JOOOsc) /lsed /f101T' symbols rbrlll ""foil', 1I1r/!.:illl!, IIIfIIlbny lIIore call/pllet bur barrier ro lellrl/ W /lse.

with extra power, forming a mathematical elite. In many cultures, numbers have been closely allied with divinity and magic, and preserving the mystery of numbers helped to maint:lin the authority of the priesthood. Even the Catholic Church was to indulge in this 10,000 jealous b'l.lardianship 54,321 == 5 X 10,000 of numbers in the 10,070 == 1 >< 10,000 European .M.iddle Ages. Other cil>hcrcd systems include Coptic, Hindu Brahmin, Hebrew, Syrian and early Arabic. Ciphered systems often use letters of the alphabet to represent numerals .

position of the numerals to show their meaning. This ean only work when there is a symbol for zero, as otherwise there is no way of distinguishing between num bers such as 14, 204 and 240, a problem encountered by the Babylonians.

GETTING INTO POSITION

dated was developed by the Sumerians from 3000 to 1OO0BC, but it was a complicated system that used both 10 and 60 as its bases. It had no zero until the 3rd century Br:, leading to ambiguity and probably confusion. Even after zero was introduced, it was never used at the end of numbers, SO it was only possible to distinguish between, say, 2 and

1,000 4 X 1,000

100 3 X 100

10 2 X 10

1 1 Xl

a x 1,000 a x 100

7 X 10

1 XO

A positional system loan show very large numbers as it does nOt need new names or symbols each time a new power of lOis reached . l1lt~ Mrli~t

Positional number syStems, such as au r own modern SYStem, depend on the position of a digit to sho\v its meaning. A positional system de\'elops from a multiplicative grouping system such as Chinese by omitting the characters that represent 10, 100 and SO on and depending only on the

positional

~y;rem

that em he

IS

STAItTlN G WITH

NU MBllt~

SUMERIANS AND BABYLONIANS The fertile area of the two

Me~opotamia,

river ~ ngri~

between

and Eu p hrates, has

been called the cradle of civilization. Now in Iraq, it was settled by the Sumerians, who

by

the

middle

of

millennium BC had established

the

fourth

perhap~

the

earliest civilization in the world. Invading Akkadians in the 23rd century BC largely adopted Sumerian culture. The period from around 2000BC to 600BC is generally called Babylonian. After this, Persian invaders took over, but again continued rather than replaced the culture of the area.

200 from the cOntext. This was sometimes easy and sometimes not. The statement '1 have 7 sons' was unlikely to be interpreted as '1 have 70 SOilS' - but a statement such as 'An army of 3 is approaching' contains dangerous ambiguity. An army of 300? No problem. An army of 3,000, or 30,000 or even 300,000 is a very different Illa tter. One of the two number systems in use in Ancient Greece, that most popular in Athens, used letters of the Greek alphabet to represent numbers, bCb>lnning with alpha for I, DNIl tor 1 and SO on up ro 9. Next, individual letters were used for multiples of ten and then for muh:iples of lOO, so that any three-digit number could be represented by three letters, any four-dib>lt numher by four letters, and so on. They didn't haw enough letters in their alphabet to make it up to 900 with this system, so some of the numerals were reprL'Sentcd by 16

archai c letters rhey no longer used for writing. For numbers over 999 they added a tick mark to the right of a letter to show rhat it must be multiplied by a factor of 1,000 (like our comma as a separator) or the letter 11111 as a subscript to show multiplication by 10,000. To distinguish numbcrs from words, they drew a bar over numbers. GrL'ek philosophers larer came up with methods of writing very large numbers, nor because they especially needed them, but to counter claims rhat larger numbcrs could nOt exist since t here waS no way of representing them. The Mayans used a complete positional system, with a zero, used thoroughly. The earliest known use of zero in a Mayan inscription is 36nc- Mayan culture was discovered - and consequently wiped out, along with the Mayan civilisation - by Spanish invaders who came to Yucatan in

W HER£ DO NUMBUS COM E nOM ?

the early 16th century. The Mayan number system was based on 5 and 20 rather than 10, and again had limitations. The first perfect positiona l system was the work of the Hindu s, who used a dot to represent a vacant position .

M~()p otami:l

THE BIRTH OF

Adding a diagonal line hetween the horizontal strokes of the Brahmi '2' and a verti ca l lin e to the right of the strokes of the Brahmi '3' m:lkes recognizable versions of our numeral~. The Brahmi numerals were part of a ciphered loystcm, with separate ~ymb()ls for 10,20,30 and SO 011.

H I NDU-ARABIC NUMBERS

ahout ;\0650 refers to nin e Hindu numbers. 2

3

= -

4

5

+

" "

6

7

1

B

...,

9

I

The numhers we use today in the \Vest have a long histOry and originated with the In dus valley civilizations more than 2,000 years ago. They are first found in early Buddhist inseri ption s. The use of a single stroke t() stand tor 'one' is intuitive and, nOt surprisingly, many cultures came up with the idea. The MOVING WESTWARDS orientation of the stroke vari es - while in The Arah writer Ibn al- Qifti (\ 172- 1242) the -\;Vest we still use the Hindu-Arabic records in his OJTOIIO/O&'Y of tbe SeiJo/tlTS how vertica l stroke. (I), the Chinese use a an Indian scholar hrought a hook to the horizontal stroke (-). But what about the .~t'.c()ll d Ahisid C:lliph Abu Ja'far Abdallah other numbt:rs? The squiggles we now use ihn Muhammad al-Mamllr (7 12-75) in to represent 2,3,4 and .~o on? Baghdad, Ira q. in 766 . The hook W:lS The earliest, 1, 4 and 6, date from at least the 3rd century Be :md are found BRAHMAGUPTA (589-668) 111 the In dia n Ashoka The Indian mathematician and astronomer Brahmagupta inscriptions (these record was born in Bhinmal in Rajasthan, northern India. He thoughts and deeds of the headed the astronomical observatory at Ujjain and Buddhist Mauryan ruler of published two texts on mathematics and astronomy. Hi s India, Ashoka the G reat. work introduced zero and rules for its use in arithmetic, 304-2328C). Th e Nana and provided a way of solving quadrati c equations Ghat inscriptions of the equivalent to the formula still used today: second century Ile added 2, 7 .\'. _h:i:..)4t1(+'? and t) to the li st, and 3 and 5 2" are found in the N asik eaves of the 1st or 2n d century AD . Brahmagupta'5 text Brohmasphufaliddhanta was used to A text written by the explain the Indian arithmetic nef'ded fo r astronomy at the Christian NestOrian bishop House of Wisdom. Severns Sebokht livin g in 17

STARTING WITH NUMBlRS

probOlbly the Bmhmaspbllfasiddbantn (The Opening of tbr U7Iiverse) written by the IndiOln mathematician Brahmagupta in 618. The caliph had founded the 1-1ouse of Wisdom, an edueJtional institute that led intellectual development in the Middle East at the time, translating Hindi and Classical Greek texts into Arabic. H ere, the BmlmltlsplJllftlsiddbtlllftl was translated into Arabic and Hindu numbers tOok their first step tOwards the \;Vcst. The diffusion of the Indian numerals throughout the 1\'liddl e East was assured by two very important texts produced at the Housc of \Visdom: 011 tbe CaJC1IJat;ofl with Hindll Numem/s by the Persian

mathematician al-Khwarizmi (c. 815), and 011 the Use of tbr Indian Numerals by the Arab Abu Yusuf Yaqllb ibn [shaq al-Kindi (830). A system of counting angles was adopted for depicting the numerals 1 to 9. It's easy to see how the Hindu numerab could he converted by the addition of joining lincs to fit this system - try counting the angles in the straight-lin e forms of the numerals we use nOw:

1Z~~Sb lB~

MUHAMMAD IBN MUSA Al·KHWARIZMI, c7B0-8S0 The Persian mathematician and astronomer al·Khwarizmi was born in Khwarizm, now Khiva in Uzbekistan, and worked at the House of Wisdom in Baghdad. He translated Hindu texts into Arabic and was responsible for the introouction of Hindu numerals into Arab mathematics. His work was later translated into latin, giving

--1"1

Europe not just the numerals and arithmetic methocls but also the word ' algorithm' derived from his name. When al·Khwarizmi's work was translated, people assumed that he had originated the new number system he promoted and it became known as 'algorism'. The algorists were those who used the Hindu·Arabic positional system. They were in conflict with the abacists, who used the system based on Roman with an abacus.

18

W H ER { 00 NUMBERS COME nOM?

A FU SS ABOUT NOTHIN G

Zero was adopted around the same time;

zero, of course, has no angles. The Arab scholars devised th e full positional system we lISC now, abandoning th e ciphers for multipl es of ten used by the Indian math ematicians. Not long after, the new fu sion o f Hindu-Arabi c number systems made il5 way to Europe through Spain, whi ch was un der Arab rul e. The earli est European tt;'xt to show the Hindu-Arabic numeral s was produced in Spain in 97 6. ROMA NS OUT!

Of course, Europe was already using a number system when the Hin du-Arabi c nOtation arrived in j\'loori sh Spain. Mter the fall of the Roman Empire in th e \Vest, tradition ally dated A04 76, Roman culture was only slowly eroded. Th e Roman num ber system was un chall cnbTCd for over 500 years. Alth ough th e Hindu -Arabi c numerals crop up in ,\ fLow works produced or copied in th e 10th century, they did not enter th e main stream for a long time. 1

I

5

V

10

X

50

L

100

C

500

0

1,000

M

5,000

(I)

10,000

(I)

50,000

(I)

100, 000

(I>

The conce pt of ze ro might seem the antithesis of counting. Wh ile zero was only an absence of items counted, it didn't need its own symbol. But it did need a symbol when positional number systems emerged. Initially, a space or a dot was used to indicate that no figu re occupied a place; the earliest preserved use of this is from the mid·2nd millennium Be in Babylon. The Mayans had a zero, represented by the shell glyph:

~ This was used from at least 368e, but had no influence on mathematics in the Old World. It may be that Meso-Americans were the first people to use a form of zero. Zero Glme to the modern world from India. The oldest known t ext to use zero is the Jain Lokavibhaaga, dated AD458. Brahmagupta wrote rules for working with zero in arith metic in his Brahmasphutasiddhanta, setting out, for instance, that a number multiplied by zero gives zero. This is the earliest known text to treat zero as a number in its own right. AI·Khwarizmi introduced zero to the Arab world. The modern name, 'zero', comes from the Arab word zephirum by way of Venetian (the language spoken in Venice, Italy). The Venetian mathematician luca Pacioli ( 1 445~1514 or 1517) produced the first European text to use zero properly. While historians do not count a 'year zero' between the years 1 Be and ADT, astronomers generally do.

\9

STARTING WITH NUM BlRS

LETTERS FROM ABROAD The Romans used written numerals before they could read

and write language. They adopted numbers from the Etruscans, who ruled Rome for around 150 years. When the Romans later conquered the Greek·speaking city of (umae, they learned to read and write. They then adapted the numerals they had taken from the Etruscans to make Roman letters.

As the Empire grew m extent and sophistication, the Romans needed larbrer and larger numbers. They developed a system of enclosing figures in a box, or three sides of a box, to show that they should be multiplied by 1,000 or 100,000. The system wasn't used consistently, though, so

fV1 could me:ln either 5,000 or 500,000. Arithmetic is virtua lly impossible with R oman numerals and this was to lead to its eventual replacement. XXXVIII +

XIX LVI!

(38 + 19 = 57)

For the purposes of accounting, taxation, census taking and so on, Roman account;lIlL~ always used Cd fur the next stage, so 1 in a square is 1 in twO nested triangles. The first nested triangle is 21 '" 4, so the next nested triangle is 4~ = 256. ® (a number /I in a pentagon) is equivalent to 'the number n inside n squares, which are all nested' . Originally, this was the limit ofStcinhaus's system and he used a circh:, for this: @.

Cd) starts from 156 Jj~ and evaluates this in the same way 256 times. Steinhaus gave the numher @ the llame a tJlCgll, and @ the name lIIogistolJ. Most'r~~ number is 1 inside a polygon with mega sides. MOVING ON

Another system, Steinh:ms-i\1oser notation, uses polygona l sh:tpe.~ to show how many times a numher muSt be raised to a power.

@] (a numher 'T/ in a square) is equi'~Jlent to 'the numher n inside n triangles. which are

Now that we are equipped with l1rgt' enough numbers, we em begin to put them to work. \iVhat numbers can (k) on their own is the suhject of pure mathematics; wh:lt they can do when they are recruited into the service of other discipline.~ is applied mathematics. A culture must develop :It iL'ast a littl e pure mathematics before it can start appl~'ing numbers Il) real-world problems such as building, economics and :lStronomy, SO we will start with number theory.

CHAPTER 2

•

".'

..... I

...-

. ,~~

,

'to

NUMBERS put to work

•

Counting is a good start, but any more sophisticated application of numbers requires calculations. The basics of arithmetic addition, subtraction, multiplicanon and division - came in to usc early on through practical applications. As soon as people started to work with numbers in this way, they began to notice patterns emerging. Numbers seem to play tricks, to h.'1VC a life of their own and to be able to surprise us with their strange properties. Some arc simple bur elegant - like the way we can muJtiply a two-digit number by II simply by adding the digits together and putting the result in the middle: 63 x 11 :: 693 (6 + 3 '" 9, put 9 between 6 and 3). Some are breathtaking in their sophistication. Number theory, which includes arithmetic, is concerned with the properties of numbers. Ancient people imbued numbers with special powers, making them the centre of mystical beliefs and magical rituals. Modem mathematicians talk of the beauty of numbers.

A

11/1/1/

/Ires

1/1/

IIbflCIIS ill fI Jllpllllere >I1:m·d shop, c.1 890.

~

/ ' NUM8ERS PUT TO WOR k

/~ Putting two and two together Thl'_ rules of arithmetic provided the ancients with methods for working out fairly simple sums, bur as the numbers involved grew larger, tools to help with and eventually to mechanize - calculation become increasingly important. Tools ro simplifY addition, subtraction, multiplication and division emerged very early on. Over the last few centuries these simple aids have nOt been sufficient and our tools for working with Ilumbers have hecome increasingly complex and technically sophisticated, until we now have computers that carry out in a fraction of a

second calculations that would have seemed quire inconceiv:Jble to the earlieSt mathcmatici:ms. STRIN GS, SHELLS AND STICKS

The earliest mathematical tOols wcrc counting aids such as tallies and beads, shelL~ or stones. The Yoruba in west Africa used cowry sbells to represent objects, always reckoning them in b'TOUPS of 5,20 or 200, for !!Xample. Other civilizations have used different objects. In lvleso-America, the Inca ci,~lization had no written number system but used khipll (or flllipll) - groups of knotted srrinbr:; - to record numbers. A khiplI consist~ of coloured strands of alpaca or llama wool, or sometimes COtton, hanging from a cord or Shdls

hac~ hel'lI

{/lid {/S C1111"fllry.

36

lISell liS colilltillg aids

A Sooin fCboolboy IISes a Rllssiall ab,wlf - fhr schocy - during tl11lfIth,·/eSSOIl still widely I/wd

ill

ill

1920. AbacllSes {IIY

Rllssifl, bur

110

umgl'r

ill

schools.

rope. 1t could be used to record ownership of goods, to calcubte and record taxt!S and census data, and to Store dates. The strings could be read by Inca acc()untanl5 called flllipuralJlayors, or 'keepers of the knots'. Different-coloured strands were apparently used to record differi ng types of information, such as details relating ro wal; taxes, land and so on.

PUTTING TWO ANO TWO TOGETHER

KNOTTY PROBLEMS

The position ot a group of knots on a khipu shows whether that group represents units, tens, hundreds, etc. Zero is indicated by a lack of knots in a particular position. Tens and powers of ten are represented by simple knots in dusters, so 30 would be shown by three simple knots in the 'tens' position. Units are represented by a long knot with a number of tums that represents the number, so a knot with seven turns shows a seve n. It's impossible to tie a long knot with one turn, so one is represented by a figure-ot-eight knot. Khipus recorded information such as population censuses or details of crops harvested and stored. Alrhollgh it lookr lif f a dfforativr jrillgr, tbr

khipu 1:.'ar a sopbisticaud new llllting aid. T bis ollr was n/(ulr ill Prru c.1.J 30- 15 32.

N orth American tribes also used knotted strings, called W01JlP01Jl, and knots in leather straps have been used in less sophisricated arrangementpl"esmred by rbe baH at rbe cmTIT of tbe flrmillary

~bf"n,

rbe apparellt O1vits of

o/bel" bodies by tbe l"illgr aI"QIlIll/ ir.

all infinite number of twin primes. That s~cms r~asona ble, as it on Iy means they don't have to run om at some point. Bm it hasn't heen prown to be true. Therc is also a 'wcak' twin primes conjecture, which has been dcmonstratcd. This states that the number of twin primes below a number x is approximately given by this horribly complicated expression:

SP{C IAL NU M SER S ANO UQU ENCU

THE GOLDBACH CONJECTURE

In 1742, the Prussian mathematician Christian Goldbach wrote a letter to the Swiss mathematician and physicist Leonhard Euler in which he set out his belief that every integer greater than 2 can be written as the sum of three primes. He considered 1 to be a prime, which mathematicians no longer accept. The conjecture has since been refi~ed and now states that every even number greater than 2 can be written as the sum of two p rimes. Goldbach could not prove his belief (which is why it is a conjecture and not a theorem), and no one has been able to prove it since. it has been verified by CDmputer for all numbers up to 1,01 8 (to April 2007), but a theoretical proof is still needed.

[~

J(logx)' ~ 1.320323631

J~

(Iogxi

Th~ll! '1:. 'I1SII" {/ IVCf/( J~{/IIJj'slI/lIl1

ill (be ferrer GQId/!(lch 'i:J1'1)rr

(Q

rtlll.:

Elllet;

Inn {blll'r hUiJ.1 ir is wir" 1ffillbofloricillllS,

propl'r di\>isors. Thi~ means that if you add together all the numhers that the numher can be divided by, the answer is the number itself. For t!Xample 6", 1 +2+3=lx2x3 28", T + 2 + 4 + 7 + 14 '" 1 X 2 X 14", 1 X 4 X 7

Don't worry about the e,x pression - it doesn't matter. VYhat is interesting to think aOOut is why it exists at all. \\!hat is it about numbers that makes it possihle ro find an Euclid first proved that the formula 1",1(2"_ 1) expression like this? Th e number in the gives an even perfect number whenever middle, 1.310313631, is called the prime conStant. It has no other known M1HA1LESC U'S THEOREM relevance except in this in 1844, Belgian mathematician Eugene Charles Catalan prediction of twin primes. (1814- 94) conjectured that 2' '" 8 and 3' '" 9 form the only example of consecutive powers (Le., 2 and 3, with cube PERFECT NUMBERS and square, 8 and 9), It was finally proven to be the case Perfect numhers are those by the Romanian mathematician Preda Mih~ilescu in 2002. which an:' the "''lllll of all their

51

2n _ I is prime . There arc currently 44 perfect numben; known, the hight:st of which is 2 J!.,tIl.b5h x (2 11·SIIl .657 -1). It has 19,616,7 I 4 digits .

o o 0 000

AMICABLE NUMBERS

Amicable numher.~ comc in p3irs. Thc proper divisors of om.' of the pair, added together, produce tht: other. The numbers 220 3nd 284 ~re an amicable pair. Tht: proper divisors of 220 arc 1,2,4,5,10, II, 20,22,44,55 and 110, which 3dd('d tOgether make 28+; and the proper divisors of 284 arc 1,2,4, 71 and 142, which tOb'Cther make 220. Pythab'Oras' followers studied amieJble numbers, from around 5OO8C, believing them to have 1l13ny mystical properties. Thabit ibn Qurrah (836-901), 3 tr3nsbt()r of Greek mnhenutlC31 tc.xt.~, discovered 3 rule for nnding amic3ble numbers. Arab m3them~ricians continued to study them, Kamal aI-Din Abu'I-H asan Muh3mmad 31-Farisi (c.1260-1310) discovering thc pair 17,926 and IH,416 and Muhammad B3qir Y.1Zdi nnJing 9,363,584 and 9,437,056 in the 17th century. POLYGONAL NUMBERS

Some numhers of dot.~. Stones, seeds or other objcct.~ can be. arr3ngeJ into regular polygons. For example, six stonl!.~ can be 3rranged into 3 perfectly regular triangle.

Six i.~ therefore :I. triangular number. If we 3dd ~n cxtra row of stoneS at t he bottom, we get the next triangular numher, ten:

o o

0

000

o

000

Nine StOnes can be arranged intO

o

0

:J.

s(]uare:

0

000 000

The nl!.xt square number has four side, giving 4! = 16.

011

e3ch

Some numbers, such as 36, 3re both triangular 3nd squ3rl': 000000 000000 000000 000000 000000 000000

o 'Six i5 a number petfeet in itself, and not because God created all things in six days; rather, the converse is true. God created all rhing5 in six da~ becau5e the number is perfect.' StAugustine (AD354 -430), The City of God

52

00 000 0000 00000 000000 0000000 00000000 P()lygon~1

numbers 3rc increased by incremcnting each side by onl' extra unit.

SP(CIAl NUMBERS AN O UQU(NCU

TRIAN GU LAR NUMBERS

[

3

6

•

o

o

••

o

[0 o

o

0

•••

o

0 00

••••

SQUARE NUMBERS

•

0

[6

9

4

[

•

••

0 0

0 0

• •

•••

0

0

0

0

0

0

0

0

0

• • •

••••

PolYb"Onal numbers have been studied since the time of Pyt hagoras and were often used as the basis of arranb>ementS for talismans. Notice how the prc\'ious triangubr or square number is incremented to form the next in the series. TRIANGULAR

SQUARE

NUMBERS

NUMBERS

1

1

3 (= 1+2)

4 (= 1+3)

6 (= 3+3) 10 (: 6+4)

9(=4+5 )

15 (= 10+5 )

25 (= 16+9)

AI/Ionio Galldl iliaR-pmlTl'li fJ1!IIlgic SqIllD? ill th~ Ciltlxt/ral rf dx Sllgmdll Fllmlfi" iI/Btl/TrlOI/a. TIx 1II11gir IIImm- is 33,

it) has three squares on each side and the magic constant is 15:

16(=9+7)

21 (= 15+6)

36 (= 25+11)

28 ('" 21 +7)

49 (= 36+13)

MAGI C SQUARES

A magic square is an arrangement of numbers in a square grid so that each horizontal, vertical and diagonal line of numbers adds lip to the same total, ca lled the mabric constant. The small est magic square (apart from a box with the figure 1 in

th~ YIIPfX1.w ~~ rfChrist at his deatll.

15

2

7

6

15

9

5

1

15

4

3

8

15

15

15

15

15

This IS known as the Lo Shu sguare after a Chinese legend recorded as early as 650BC This tells how \,illagcrs tried to appease the spirit of the flooding rivcr Lo and a turtle came our of the watcr with markings on its back that depicted the magic square. The people were able to use the pattern to control the river.

Magic squares have been known for arowld 4,000 years. Th!..')' are recorded in ancient Eb'YPt and Lldia and have. been attributed with special powers by cultures around the world. The first known magic squares with fin' and six numbers on L'11ch side arc described in an Arab [ext, the RflJa 'il Ib~"Wml fll-S(/fo (E1lt),dopedia of tbe Brefbrt'll of Pllrity), written in Baghdad around 983. The tirst European to write about magic squares was the Greek Byzantine scholar Manuel .M oschopoulos, in 1300. The Italian mathematician LUL-a Pacinli, who recorded the system of double-cuny book-keeping in 1494, collected and smdied Illabrie squares. (He also compiled a treatise on numher puzzles and magic that lay undiscovered in the archives of the University of Bologna until it was published in 2008.)

PI A~

well as numbers that foml series or pattems, there are several strange and significant single numbers. The fir.~t to he discm'ercd W;lS pi, n . This ddines the ratio of a cirde's diameter to its cireum ference, ~o that the circumference is nd

where d is the diame.ter. Th e. vJlue of rr is a decimal number with an infinite number of digits after the decimal poinL It IH.'gins 3.14159 (which is a good enough approximation for most purposes). That the ratio ofthe diameter of a circle to irs circumference is always the same has been known for SO long that iL~ oribrins can't be traced. The Eb'YPtian Ames Papyrus,

c.l650BC, uses a value. of 4 X (819f = 3.16 for 11". In the Bible, measurements relating to the building and cLJuipping of the temple of Solomon, c. t)50nc, use a value of 3 for 1L The first theoretical calculation seems to h:tve been carried Out by Archimedes of Syracuse (287-212nc). He obtained the approximation

He knew thar he did not ha vc an. accurate value, but the average of his twO bounJs is 3.1418, an error of a bout 0.0001. Later mathematicians have refined the approximation by discovering: more decimal places.

e Another strange and very significant number is e. The value of c was first discovered by Jakoh Bernoull i, who tried to discover the value of the expression lilll

•

""

(I . . 1)' II

while working on calculating compound interest. \-Vhen evaluated, t he. expression gives th e series that defines e.. The tI.rst known use of the eon Stant, represented by the letter Il, is in letters from Gottfried Leibniz to Christi:lan Huygcns written in 1690 and 1691. Leonhard Euler was the first to use the letter e for it in 1727, ~nd the fir.') t published use of e was in 1736. He possibly chose I.' as it is the first letter of the. word 'exponential'. I.' has an infinite. numher of digiL~ after the decimal place, as it is defined (among

SP{ CIAl NU M 8[RS ANO UQUENCU

othcr methods) as the sum of allnumbcrs in an infinite series - see panel. page 41, UNREAl!

The imaginary number, i, is detined as the square root of minus 1. The term imaginary numher was used by the French philosopher and mathematician Rent! Descartes (1596-1650) as a derogatory term, but now mcans a numher that involves the imaginary square rOOt of -I :

en ,+ T '" O. This, known as Euler's identity, is a special case of a rule which rdates complex numbers and trigonometri c functions.

(A negative number t::J.n't 'really' have a square rOOt as when a number is squared, whether it is positive or negative to Start with, it always gives a positive result.) A complex numher z is defined as z'" x + iy

where x and y afe ordinary numben:. Imaginary and complex numbers were encountered first in the 16th cenrury by Gerolamo Cardano and Niccolo Tartaglia while investigating the roots of cubic and quartic equations, and were first described by Rafael Bombelli in 1572. H owever, even negative numbers were distrusted at the time, so people had littl e time for imaginary numbers. It was in the 18th century that it began to he taken more seriously. It was brought to thc attention of mathematicians properly by Carl Friedrich Gauss in 1832. Strangely, the special numhers come together in the expression which has been call ed the most startling in the whole of mathem:uics:

Tb .. Grnk ffltllbcmoridllll Pyrhogol'llrdl'1noll#rarrs

bir rb..o/'~m 1)1/

1)1/

rbt grolmd,

dgbr'llIIg'.-d "'71l11ghs by dl'tr;1'illg

,;S-/ '1 NUM 8 US

PUT TO WO Rt(

1'/ Unspeakable numbers The t'oncept of banning a number may seem bizarre, but it has happened for millennia and still happens even tOday. Some numbers havl' been considered just too difficult or dangerous to countenance and havl' been outlawed by rulers or mathematicians. But a lwnned number dOCiin't go away, it just goes underground for a while.

'It is rightly disputed whether irrational numbers are true numbers or false. Because

in studying geometrical figures, where rational numbetJ desert us, irrationals take their place, and show predsefy what rational numbers are unable to show.. we Gre moved and compel/ed to admit that they are correct .. Michael Stifel, German mathemat ician

(1487-1 567) PYTHAGORAS' NUMBER PURGE

The ancient Greek mathematician Pythagoras did not recognize irrational numhers and banned consideration of negative numhers in his School. (An irrational numher: is one that eannot be expressed as a ratio of whole numbers; so 0.75 is a rational numher as it is ;/. hut j"( is irrational.) Pythagoras had to aclmowledb't! that his ban caused problems. His theorem, which finds the 1t~nbrt.h of a side in a rightangled triangle from the lengths of the other two sides, insrantly runs into problems if only rational numbers are recognized. The length of the hypotenu~ (longt!st side) of a right-angled triangle with two sides one unit long is the square rOOt of two - an irrational number ("" 1.414).

Pythagoras was una hIe to prove by logic that irrational numbers did not exist, hut when H ippasus of Metapontum (born (.500Be) demonstrated that the square root of 2 is irrational and argued for their existence, it is sai d that Pythagoras had hi m 56

drowned. According to legend, Hippasus demonstrated his discovery on board ship, which turned out to have been unwise and the Pythagoreans threQmetry that algebraic questions first surfaced. Early on, specific, practical problems in algebra were neither systematized nor represented in a way which we would now recognize as algebra yet they provide th e origins of algebra as it was later fOnlmlated. FIELDS AND CELLARS

Babylonian clay tablets III the British Museum include a number of problems which would now be formulated as quadratic or cubic equations. These rdate to building projects and involve working with areas and \'Olumes. Some problt:ms related IU dividing up an area in partS with different proportions. 1t is easy to see how a problem in area can lead to a quadratic equation.

, , 2

Here the area of the larger (enclosing) rectangle is (a + 2)(a + 1) '" a' + 3a + 2

Similarly, cubic equations can be derived from Babylonian problems relating to digging cellars. The earliest known attempt 122

The lIlnhod for solvillg rrmllltfllll'oflS

N11/an011S

IIlnlled aftl'l' Carl Friedrich Gauss bad bel'll tlsed

;11 rhl' East 2.000 JtYlrs ulrlin: to write and tackle cubi c equations is in the form of 36 problems about construction in a clay tablet nearly 4,000 years old. Such problems were expressed in words by both the Babylonians and Egyptians, and by mathematicians for many centuries afterwards - for example 'the len gth of a room is the sa me as it.~ width plus 1 cubit; it.~ height is the same as its length le~"S 1 cubit.' The Babylonians did not attempt any general rul~ or methods of treatment for problems of these types. They dealt only with the specifies of each problem and seem to have had no grasp of a general algorithm that could help them solve all problems of a simi lar type. The Ancient Egyptians, tOO,

ALGEBRA I N TH( ANCI£NT WOR lO

solved practical problems that would now be exprcssed as linear or quadratic equations, but again without rcC()urse to any formal notation and without recognizing them as equations. The Chinese text The Nille OJllpfers (2nd to bt cenmry Be) includes a chapter on soh~ng simulcmeous linear equations for two to seven unknowns. They wen' solved using a counting board or :;l.Irnce and could include neg'ative coefficients. The description of c'luations with negative coefficients is the earli e.~t known use of negative numbers. The method used is now known in the \Ve~t as Gmssiall elimination after Carl Friedrich G:I\.l~S who used it 2,000 years later.

yielded more than one solution he sropped after arriving at the first - (..'Ven if there were an infinite llumber of solutions (as for an equation of the type x - y '" 3). H e developed a method for representing equations which was less cumbersome than writing them um in wurds, bur was ~rill nOt comparable with modern methods. A~ the Greck~ used thi:' letters of their alphab~t for numbers, thert: were no recognizable ,"ymbuL~ immediately ava.ilahle to represent varia bles. \Ve ean usc x, y, a, b, m, n and so 011. to stand for variables and mmtants because we have separate symbo ls for numbers, and SO an e:"llression such as 2x is unamhiguous. Diophanrus adopted some variants on Greek letters, and used symhols FROM GEOMf.TRYTOWARDS ALGEBRA to indicate squaring and cubing . His ~YS[CIl] 1"n the middle of thc 3rd century AD, the of ablm..,'Viations was:m intermediary st:lge Hellenistic mathematician Diophanrus of between the purely discursive explanation Alexandria developed new methods for uf pro hi ems and the purely symholic in usc so h~n g proLk'IJl.~ that would now be sho ....'T\ now. It also g"Jve him the opportunity, not as linear and 'luadtatic equations. H.is work, .~een ur exploited before, of dealing in ArifbllletiCtl (of which only part has higher power.~ than mbl'.s. Some of his survived), cOIlt'Jins a number of algebraic problems include a notation that means cquations and methods for solving them. 'square-square' or 'cube-cube', indicating Diophantus applied his methods ro the powers of 4 and 9 respl.'Ctively. problems in hand, but did not e.xtend them 1n addition, Diophanrus had no concept to general solutions. Like the earlier of an equality - of twO balanced expressions Greeks, hc dismissed any solutions that between which parts could be moved or on were less than zero, ami whcn an equation which idcntic-al operations could be carried (JUt. Nor did Di ophantu s deal with more than one INDIAN QUADRATI CS unknown at a time. H i:' An ancient Indian text, one of the Sul ba sutras written by always sought a way to convert a sC(:(md unknown Baudhayana around the 8th century Be, fi rst cites and then into a.n expression built solves quadra tic equations of the form ax' '" c and axl + bx = c. Tht>..'ie occurred in the context of building altars, and around the first. So, for 50 relate to a practi[al problem in three dimensions. example, in a problem that ca ll s for two numbers whose

n'"'"" " ~OR M UtA

sum is 10 and the sum of whose squares is 108. D iophantlls would not write, as we may. x + y = 10; x1 + y.' '" 108, but might tCflll them (x + IU) and (x - 10), the second equation then becoming (x + lOy +(x - 1W :20S.

ORDERS OF EQUATION

Polynomial equations are those that contain a series of terms, each of which has a variable raised to any power, multiplied by a constant (ordinary number). For example in the following equation

D I OPHANTINE EQUATION S

Diophantine equations arc those III which all the numh~rs involved, including those in the solutions, are whole numbers (which can be positive Of negative) . They fall into three categories: rhost': with no solution, rhose with a fixed numher of solutions and those with infinitely many solutions. For e.x:lmple. the equation 2x+2y:l

has no solutions, because ml'n"! arc no values for x and y that are whole numbers

the first term consists of "II! X 1, the second 01 Xl X 2 and the last the constant B (or XO X .8). Mathematicians refer to polynomial equations as bei ng of the first order, se(ond order and so on depending on the highest power they contain. So a quadratic equation such as that above is called a second-arder equation; an equation including a cubed term (x~ is a third· order equation.

4x + 6y '" 24

that can give the amwer t (the sum of two

even numbers is always even). The equation x - }' = 7 has illfinitel~' many solutions as we can continue to pick larger and larger values of x and y. The equation 4x '" S has only one solution: x = 2. Diophantine equations ;Ire useful for dealing with qu:mtities of obje('t~ that cannot be divided - such as numbers of p(.'Ople. So, for instance, if there is a choice of (~ ars to rake 24 people on ~ trip, some of whieh c~rry four and some of which carry six passengers, and ~ll must be full, we could write ~ Dioph~ntine equation, since the only useful solutions assib'll whol", numbers of people to whole numbers of cars: ,2
uring, between arithmetic and gl'ometry, as wl'll as the problems of the continuous and thl' discrete at the heart of Zcno's paradoxe.'i.) Think about any system that involvcs continuou~ change - water flowin g over a dam or air over an aircraft wing, for example. As local conditions vary, the rate of flow is not conStant. l\ 'l easuring it at any moment involves some kind of approxim:nion or averaging as the time interval could always he made small er. Only by freezing time could we take an accurate measurement. But fl ow depends on time. so if we freeze time the flow is zero. 1t is not only rime that can he endlessly subdivided. For c.xample, as tl'mperature chanf,'"Cs from 1" to 3", it must go through aJl infinite numher of imermnliate stages; even 1" and 3° thcmselvcs are infinite decimals, with an infinin.· numher of zeroes aftcr the decimal point. In modelling continuous change, we must deal with these fleering values - and th ey arc necessarily infinite decimals. The concepts and deductive stl1.lctures hehind infinite quantities came to preoccupy mathematicians working with calculus as they struggled to develop ri gour. For analysis to become a rigorous and dependable tool, mathematicians first needed to tind some way of dealing with thc vagueness of these ghosts of quantities and moments. The Gernlan mathematician Karl Weierstrass (1815- 97) was the first to produce a complct~ly satisfactory definition of the limit o f a seri es. H e became known as

CAlCUlUS ANO BHONO

the father of modern analysis for devising a test fur the convergence of series and fur his work on functions. Using the sequence

INFINITE SERIE S

An infinite series is a series with an infinite number of terms. For example,

\Veitmtrass would say that all we need to do is pick the level of error (or approximation) that is acceptable (~) and then continue with the series until we reach a term lIn which is smaller than the error, then we C3 n say th3t the series has reached its limit. Thi s the need for nebulous removes infinircsimals and gives a real numher which satisfies the requirements. Also, although the series 3pproaches its limit, it docs not have to reach it for Weierstrass's condition to be met. Now, the margin of approximation could he stHed 3nd the

is an infinite series, with each term being half of the last. The limit of this series - t he number that would be reached if we could get to the end of the infinite number of terms - is 1. Because the series reaches a definite limit it is said to converge. Other series do not converge, such as

This series diverges as it never settles to a limit. Some convergent series can be ambiguous: 0+1-1+1-1+

oscillates between 1 and O.

degree of accur3CY quantified . There was no need to worry about quantities that h3d to disappear from existence - analysis was put on to a logical footing.

Gt'1"mIl1l1l1arhemllticillll KRrllf/(ierstll1J:f ,ray

rom:lTlled 7.:.'itb elimil/tltillg illCl)/lsisrmries ill edirt/IIIS (111£1 defillil/g rbe limir of II series.

CALCULUS BECOMES ALGE.BRAIC Durin g the 18th century, calculus moved 3W3Y fi-om itS geometric roots in the work of NewtOn and Leib niz and became increasin gly algcbr3ic. Geometric curves became less important 3nd algebraic functions moved to centre sta ge. Soon, complex numbers moved in on the scene. Differentiation offers a useful tool for 159

GRASP ING T HE INf IN ITE

finding local m3ximum and minimum values benvecn upper and lower limits. If we draw a curve of a function, the slope approaching a maximum point flattens out; the curve is momentarily tlat (has a slope of 0) at the maximum point, then it curves downward again, its slope reversing. As rate of change is equivalent to a tangent draWll to the curvc, it is ea!>y to Spot maximulll or minimum points - it is those pinel's at which the curve has a slope of zero and tbcn reverses its slope. This knowlcdb'C makes it possible to find the changes of direction all local maXlllllllll and mlllllllUlll points benvecll bowldarie5 without drawing the graph. Where the function differentiates to zero, the tangent to the curve is parallel tc) the axi~. Differentiation is also useful for working with :dl of the many phenomena which cxhibit exponenti:tl growtb or decay - such as population b'TUwth, or radioactive decay. By examining the rate of change at given moments, it's po~iblc to extrapolatc to find valucs for the furure (or past). WAVE FUNCTION S

The ability of calculus detcrmine ma.Xllna nlllllllla has made especially valuable for working with all kinds of waveform, from acoustics to optics, from clectromagnetism to seismic activity. The earlie;t work in this field was carried our by the English mathematician Bl"Ook T.1ylor (1685-1731) who produced 160

a mathematical description of the vibrational frcquent), of a violin string in 17 14. The French mathematician Jean Le Rond d'AJcmhcn (1717-83) refined the model in 1746 to take account of more conditions and limit'i, and of v,;uiation in some pmpenies alon g the length of the stl"ing. His dClllonStl"ation had twO wavcfonllS travelling in different dircctions. The Scottish physicistJamcs ClerkMa.nveil (1831-79) found the same thrccdimensional wave when exploring electromagnctism. It enabled him to predict the cxisrence of radio waves. Radio, television and radar are all dL·vdopments dependent on the early analyric work on the waveforms of musical inso·unlenl5. Further work on thc propagation of sound hy th e Swiss mathematician Leonhard Eulcr found a trigonomctric scries:lt the heart of the problem (1748), In 1822 the French mathematician J()seph Fouricr (1768-1 ~30) also found a trigonometric series ddining the way heat spreads along a mctal rod. From this hc developed Fourier analysis, which enabled him ro find the values necdcd to model heat spread for any initial temperature distribution. Fo uricr analysis is used to analyse complex, composite waveforms, brcaking them do wn into their component..:; and values. For instance, an audio signal can be amlyzed into Nidmrrmnl 'Drrfty' at «hool ill Edillburgb. .1f111l1'f Nffl,l"U.'tll p/'odllml l];/JI ·k

to rivrr/lllly

gt"f.'ill

pbyridsr.

CAlCUlUS AN O BUON O

LEONHARD EULER ( 170 7-83)

The Swiss mathematician and physicist

than any other mathematician has ever done,

Leonhard Euler spent most of his life in

his work filling 60 to 80 volu mes. He worked in

Germany and Russia. He published more

many fields, making significant breakthroughs not just in analysis, but in graph theory, number theory, calculus, logic and several branches of physics.blished much of the notation used now, including {(x) for a function of x, the notation fo r the trigonometric functions, the use of the symbols e (e is sometimes i and

called

Euler's

number)

and

L (for summations). He also popularized

(but did not originate) the use of the Greek letter

Jr.

His most startling discovery was

Euler's identity. 1n e + 1 '" O.

iL~ different frequenci es and amplitudes. Although his methods were not rigo rous, they were later refined and are, in essence, used today - ror compre~ing sound into downloadable At1P3s, tor example.

Too HARD

Some problems proved intractabl(! even with th e usc of calculus. The movement of the planeL~ in the solar syStem, for t;!XOlmple, is toO complex to be accountcd for by straigh tforward series. The field of dynamic system th eory has developed to ta ckle such problems. Esscntially, local data drawn from particular sit(!s within a much larger tleld arc analyzed and reslllts from these arc applied to known global properties of

Tbl' 'Kdlligwcrg bridges' problem

';l}//S

solved by

Elller ill 1736. Ir ash ;J.· babn· il is porsible 10 crors ellcb ofrbl'si'lJt7I bridges ill Kdl/igsbrrg Ollly OIlU, 1l'lIIl7Iillg to

(be Hllrlillg paim. Ettl,.r pl"Ovl'd rbllr ir

is 11(1(, dl'fillillg rbl' Elflniall prllb ill rbe PI"f)Ct'ss (II pillb (bar follU"J.JS etUb edgr, bllt ollly Ollce). His proof is rbf jirst Ibeol"e," ofgl"flpb Ibl'o'1. 161

GRASP IN G T HE INfiNITE

Kil/g Orefl/" /1 wbo affired a

GREENHOUSE GASES

Fourier was the first person to suggest, in 1827, that gases in the atmosphere may lead to increasing temperature on a planet - the greenhouse effect.

the whol e system . Today, computers analyze, approximate and ass(;!Ss solutions created in this way. Dynamic systems theory was fi rst developed by Henri Poincare (1854-1911) for a competition. [n 1885, King Oscar II of Sweden and Nonvay offered a prize to detcmlin(;! the stability of the solar ~ystem - saying whether it would continue in much the same state or whether, for e.'l:ample, a planet could fly off Oil a rob'lIt! journey of its own , perh~ps colliding with the sun. Hl'llri Pail/cOli v.}as blessM witb a formidable

1f1171/oI)

f/lld was ablf to master mfllly disaplillfs.

162

prize fol· detfl""II/illillg

b(r.l.1

Stftble fbe mla/" rySWII 'I:.'as.

NewtOn had used the inverse square law of gravitation to demonstrate the elliptical orbit of planets th~t Kepler had noticed, but he also fOlUld that the syStem was too complex to calculate if more thall two bodies were involved. The king now wanted a solution involving nine bodies - the sun and the eight planets known at the time. Poincare's solution did not, in fact, deal with nine bodies. H e restricted himself to three and even then assumed that one had negligible mass (and so negligibl e gravitation~l effect). H e modelled ~ sample of what may happen in a limited arl'a - where the path of 3 planet intersected with this are3 - and extr3polared the rate of change to come up with a prediction for the stability of the whole system. Although Poinc3f(! WOll the prize for his partial solution, he noticed a mistake in his solution and spent more than the prize money III reprinting his solution. From the end of the 1Hth century, mathematicians were more willing to accept

CALCULUS ANO BUONO

, Polnu.r~

.octlon

The hllle dire rl'[ln'fi'llts

(11/

tina

ill

poil/ty of imnwcrioll wit/; rbe

trajectory

discovered that many functions cou ld not he integrated, or hehaved in a The blind Belgian physicist Joseph Plateau (1801-83) studied the fi lms and bubbles created by soap solution. bizarre way if integrated. A~ Soap solution forms minimal surfaces- the minimal surface a consequence, integration area that can cover a space. Minimal surface mathematics was redefined by the French mathematician H enri-Leon is a productive area of research. The West German pavilion Lebesgue (1875- 1941 ) around at the Expo 1967 World's Fair in Montreal, designed by Frei Otto, was based on minimal surfa ce studies of soap films. 19CX1. Instead of taking thin slices of the b'nph vertically beneath the curve, Lebesgue complex numbers and Gauss began suggested taking thin slices horizontally. applying the principles of analysis to them This gn.'arly increased the usefulness of in 1811. Analysis using cumplex numbers - inreb'Tal calculus as it could now be used complex analysis - is possible because with discontinuous functions. It expanded com pie-x numb ers arc deemed to follow the possible applications of Fourier analysis. many of the same laws as real numbers. There are very many different brJncht.'s Modern analysis differs in many regards and applications of analysis and they spi ll from ea rly analysis . Mathematicians over into all areas of scienc£'. SOAP BUBBLES AND ARCHITECTURE

TradiUnoollntegrollon. taking """ Il1S and a listing of every item of prnperry in the land . It was written up in the Domesda), Book - a m,1ssivl' undertaking for the 11 th century and one which still providcs valuable statistics for historians. Thereafter, there '\~JS no enthusiasm for regular census-taking. Although bishops in many parts of Europe were supposed to keep count of the families- in their dioct::scs, there was littl e information about popu lati on levels. Some people even believed that taking a cemus wa s sacrilegious, citing a Story !Tom the Bible in which King D~vid attempted a census which was interrupted hy a terrible plague and never completed. The fir.~ t regular census in modem times was carried Out in Quehec, Canada in 1666. In EurQpe, fccland was the first in 1703 , followed hy Sweden in 1i49. The US held its firH ten-yearly ccnsus in 1790 ~nd the UK in 1801; the US had jlL~t under 4 million inhabimnts and the UK JO million (previous estimates had put the UK population at between 8 and 11 millio n). THE RISE OF STATISTICS

In 1662, the English Statistician J ohn Graunt published a set of .~ tatistics drawn from mortality records in London, and in the 1680s the political economist \"lilliam Petty publi shed a series of essays on 'politica l arithmetic' whieh provided smtistical records with calculations - some 175

NUMBERS AT WORK AN D PlAY

THE CENS US AND COMP UTERS

The demands of census.taking were a

considerable

spur

to

the

development of technological aids to calculating. The first machine for working with census data was used in

1870.

Census

data

were

transcribed on to a rolling paper tape displayed through a small window. In 1884 Herman Hollerith (1860- 1929)

acquired

the

first

patent for storing data on punched cards and organized the health records for Baltimore, Maryland, New York City and New lersey, which won him the contract to tabulate the 1890 census. The huge success of this census opened other markets

to

Hollerith

and

his

machines wen> used in Europe and Russia.

He

incorporated

his

Tabulating Machine Company in 1896, which later became IBM.

HQllmTb prodllced II

7IlUbllllicnl

Tahll/IITQ1· bllml

SOC IETY IS TO BLAME

QII Tbr: idf"ll

The Belgi~n m~them~tician Adolphe Quetclet (1796-1874) was :1 champion of st~tisti cs as the basis of the social study which hi:' tcmled 'social physics'. He examined d~t~ of all kinds, using the techniques common in som .. scientific di~cipline.~ of amassing a vaSt collection of dam ~nd looking for emergent p~trems. To his surprise, he found them everywhere, not JUSt in the ar(.'as where Divine Providenl'e

rbllr 1111 pnl"fJ1111/ dllrt! C01l1d be cMrd

111i1llt"I"icnlly.

176

quite bizarre, such as the monetary value of all people in Ireland. On the whole, governments encouraged or financed statistical survey.~ ~nd gu~rded the rL'Sults jc~ lous ly, using them to increase the power of the st~te. They were still inextricably tied up with ~uperstition and followed vcry unscientific methods. One of th .. most f~mous 'p(Jlitic~1 ~rithmeticiaIlS' w~s the Prussiall J()h~nn SlSsmi1ch, who published three volumes over mOre than twenty ye~rs, ending: in 1765, proving again the existenee of God revealed in the harmony of socia l stati stics. Other SL1tistics were collected by scientist.~, profession~ls of different types and hum~nitarians. Lldeed, there was a growing enthusiasm for statistics, which became something of a mania during the early 19th century. Suddenly, everything was studied, counted, audited - the weather, ~6'Tieulture, population muveml'nts, tht.: tides, the land, the Earth's magnetism ... The European countries that h~d l'mpircs surveyed their new ~l'quisitions and took cenSUSL'S in their colonies. As Americans moved westward, claiming morc land. they charted it and logSrcd its resources.

SAM pu

s AND

STATISTI CS

Tarullno %,000 Malo)ar",la~eu

Wi,ma 55,000

67,000

";=='~'~=,:30 mi

12,000

Mohliow

Ntmun /IN r

o

MI",k

"

'''m

'.~~~~~.'

_10

_ I~

_11 -

_10'

_lO'

_30'

l_per"W,e -C

_]O' Oct6 _16 ' O::\. 7

·1'"

.10' Sepl. 23 Oct , I

mi ght be expected to operate. In particular, he was impressed to find th,u crime figure s follow ed a predictable pattern. He conjectured that they are a product of society rather than indi,~duals and that, while an individual criminal may be able to rt'sist the urge to commit a crime, the overall pattern of crime rates is altered little by individual actions. He felt that the proper study was of crime ratt's rather than criminals and that the proper remedy to c rim e lay in social action, including education and an improved judicial ~ystem. Careful use o f statistics to examine the effects of changes and suggest directions for future change would, he felt sure, produce the desired results. Quete1et's thesis promptt'd some debate on the apparent contlict bet\vcen statistics and the doctrint' of frce will - if crime rates can bc deternlined by statistical methods and arc unchanging over time, how much freedom do individual s really havc over their action s?

_21 ' Sept 14 _9 ' Sept. 9

0 ' Aug. 18

0111' if the mtJSr l/{'cowplished gmphim/ J'tpl'l!wlltariml:f ofrtlltirticr nm"ntndl' is (vurles .~1illaJ'd 'r gmph of Napole(JII} disasN'o/ts campaigll ill RI/Ssill ill 1812, Ir shlTiJ.'s /l/olTality

011

the 1l-'atures for the wash L'),cle from the quantity of washing SENDAI SUBWAY and how dirty it is, for In 1988 Hitachi produced a fuzzy logic system to run example. It will calculate subway trains in Sendai, Japan. The trains need only iI optimum amounts of snap conductor and no driver. The fuzzy logic .system controls and water, the best acceleration, cruise speed and braking, taking into account temperature and the Itngth SiI(ety, comfort, fuel efficiency and the need to stop of wa.~h required. The first accurately at target positions (station platforms). system controlled by fuzzy by logic was created Ebrahim A1all1dani and Seto Assilian at refuted diagnosis can thell be fed back into Queen .Mary College, London in tbe early the s},stt'm to improve irs future t 970s. They wrote a set of heuristic rules performance. for eomrolling tbe opm-ation of a small Sets - fuzzy and classical - have steam t:Ilgine and boiler, then used fuzzy redefined mathematics fO I" the 10th and 11 St setS to c{)nvert the rules intO an algorithm to centuries. In SOllle ways, they have allowed control the system. The first commercial mathematics to be divorced from the real use of a fuzz.y ~)'stCIl] was to control a world. Higher set tbeory deals not with cement factOry in Copenhagen, Denlll:lI"k in numhers or OhjecL~ in the wurld, but with 1980. Exploration and uses of fuzzy logic concepts and relations bt'tween cuncepts. increased massively in the 19)30s, especially YCt in accommodating the imprecision and in Japan. contingenL), of the real world set theory, Fuzzy logic is not only used in control like &accds, it acknowlcdbFt'S the 'roughne.'iS' but also III expert systems, artificial of the rcal world and provide.~ a more intelligence and ~lpplicatio ns such as voice accurate (if messier) model of reality than recognition and image processing softwarc. earlier mathematics. It tries to minimize the human intervention needed in a systt'm by approximating MOVIN G ON human judgel1lenL For this it needs an Set theory works '.\';th mathematil-s far expert human to set up the rules on which divorced from numbers. A~ it docs so, it judgements are uased. uut intelligent IlL-tomes increasingly dependent upon systems t':111 then improve themselves logic. Although it may seem that logic has learning from adjustments an operatOr hcen at the heart of mathematics from the makes to the settings that the system start - after all, Euclid attL'mprcd to deri,,(chooses. [n diagnostic medicine, for all geumetry througb a sequence of logical example, 3 fuzzy systcm can look at all the steps - the applicatiun of logic was neither symptoms reportcd or monitOred in a rigorous nor closely examined until the 19tb patient and assess the likdihood of different century. Set thL'Ory, it tunled out, could he diagnoses based on tht' deb'Tt'e to which applied to devc.loping the logic needed to t:'aeh symptom is preSt'nt. A confirmt'd or give mathematics firm foundations.

oy

193

CHAPTER 9

PROVING IT

As in law, evetything in mathematics must be proven before it is accepted as true. Even the most blatantly obvious ' facts ' arc not accepted as facts unless a mathematician can provide a rigorous proof. It is not enough to put one apple with another apple to show that one plus one equals two: it must be proven that one plus one always equals two, that there are no cases in which one and one might make one , or zero, or three, or 1.7453. Often, it is much harder to prove something than to discover it and decide that it is almost certainly true. Sometimes, it takes many centuries for a theorem to be proved, as in the case of Fermat's Last Theorem. But it is the proof that defines a theorem - it must be possible to demonstrate its truth through a line of logical reasoning from axioms and other established theorems.

Tbar

rb~ filII

bar ah~}ayf

,·isell

ir

I/O

proofir ,rill rire tomorl"/JW.

P ROV I NG IT

Problems and proofs It took Jakob Bernoulli 20 years to prove that tossing a coin a large number of times will give close to a 50:50 split between hl':Jds and tails - yet as he pointed Out, th e result is olwious to anyone. \Vhy did he hother? And why did it rake So long? Although the Ancient Egyptians and Babylonians were content to work with specific examples :lnd problems, tht Greeks moved towards theorems ~nd axioms that muld be aJlplied uni\'crsally - they demanded proof. Proving th~t ~n idea holds true requires some kind of logical theoretical treamu:nt, since it is [JOt possible to try Out all Jlossible eases - to test Pythab'oras' theorem for all possihle rightangled triangles, for e.x~mp le. Proof~ aim to find fruitful relations hips between mathematical statements ~nd objects. For this reason, even theorems wbich h~vc been adequately proven in the paSt - slich as Pytha gor~s' tbeorem - may be proven anew, opening up fresh avenue,>:; for exploration. Over time, simpler proo(~ ~re di~covercd and the earli er, often cumbersome, proo f can be replaced. Many developments in mathematics came about as the result of people tcsting and Dying to prove theorems and axioms and even doubting long-h eld beliefu. The dispute over Euclid's fifth poStulate, for in~tance. was th e spur to much of the

'Ead! problem that I solved became a rule which served afterwards to solve other problems.'

Rene Descartes

progress made in geometry and ultimately the emergence of new, non-Euclidean geometries in the 19th century. Rigour in m~thematical proof increased at the end of tbe 19th ct'ntury when mathematics and logic t~me together. A sy~1:elll~tic nOtation for logic came to be used by marhematicians and some philosophers. The development of se t theory required a method of representing: logical relationships ~nd a way of dealing with concepts which did not necessarily involve any numbers ~t alL Set thec.lry even becaml' a useful ml'aIlS of demon.strating matJlL"m~ticaJ theorems. to

UN BELIEVA BLE PR O OFS

A famous problem th~t produces a proof which many people find hard to accept is the Monty Hall paradox. Named after the host of a US game show, it goes like this: Suppose (/ gtJme s};ow host shows yOIl thn:c doors. Behil/d 11UO of thrill tho·r is II golft;

behilld the /ost there is illvites

yOIl

to pick

a etll".

11 dQQI:

Tbe horl

He will tbfIJ

open nl/other door, rt'vrnlil1g (/ gont , ([lid 'Nobody blames a mathematician if the first proof of a new theorem ;! dum!y. '

Paul Erdos

giVl' you the chnllc/'. to change YOl/r cboice. IYill YOllr dUll/ITS of7ll illlli1lg be impmved ifYOII S1J,·itc/J doors? (rbe problem nSSllmes tlmt YOII would rntber hove tI {tn· tl}(ln n gont)

196

PR O BLEMS ANO PR OOfS

Most people say their chances of getting a car are unaffected if they switch doors. Mathematicians say that the chance of getting the car is increased if you switch doors: you had a 1 in 3 chance of choosing a car, 3nd this is unaffeCted by the opt'ning of another door; the chance that you chosc COrrL-ctly is still 1 in 3. If you switch, you 3rc making a new choice, where the chances arc I in 2. Switching will get you a C3f 50 per cent of the time, bur st3ying with the first door will yidd a C3r only 33 per cent of the time. The logic is easier IP follow if you think of 1,000 duors with goats behind 999 of them. Your chances of picking the door with the car the first rime are 1 in 1,000. After 998 b'03ts have llCen released IP run amok, there is a 1 in 2 chance that the other door hides the car. The obvious objection here is that there must 31so be 3 1 in 2 chance that the original door hides the. car, since prohabilitit!S must add IIp t(J one. The trick is that the problem is nOt as it appears. Your choice i~ r3ndom, hut the host IWOWJ where the C3r is. If the host randomly opened doors, coincidentally picking those that conce31ed goats, the ch3llce of finding a car 3t the end would be the same as the chance of finding a goat, whether or nOt you switched doors. The proof of this problem uses mathematical notation to show probabilities and break~ it down into small , logical Steps which naturally follow one from 3nother. This is how mathCIll3tici:ln.~ now demonstrate truth. But it has not always been the case.

Tradition maintains that Thales pro\'ed th3t the angles 3t the base of an isosceles triangle are equal, that 3 diameter CutS a circle inll) two equal parts, that opposite angles formed h}' two intersecting line.5 arc equal 3nd tha two triangles arc identical if any twO angles and one ~ide arc t:qual Sint:t' nonc of Thall'S' writings survives, it is impossihle to S3Y whether he really produced rigorous proofs of these theorems. Around fifty years latcr, Pyrh3goras proved his theorem for right-angled triangles. Since the time of Thales and Pythagoras, the basis of proof In mathematics has been to derive more complicated Statements from faCtS which are apparently simpler (though they may not actually be simpler). Genemlly, anything 1ll ge(.lllu~ try that C3n be demonstrated in clear, logic31 steps from Euclid '.~ postulates counL5 3S proven, for instance. Bllt this does not mean that a new idea is deduced first from the existing f.1Cts. Mathematicians commonly h3vc the idea

DEDUCTIVE PROOF THAT 1 '" 2

let a '" b. So it follows that

a' '" ab a1 +a' =a' +ab 2a' =a ' + ab 2a' - 2ab=a ' + ab _ 2ab 2a' _ 2ab", a'_ ab This ciln be rewritten as 2(a' _ ab) '" 1 (a ' _ ab)

EARLY PROOFS

Dividing bot h sides by a' - ab gives

The earliest known mathem3tical proofS arc said to have been provided by Thalcs.

2=1

197

P ROVIN G IT

first - perhaps as an intuition, or as something suggt'srcd by the re~1Jlts of an experimcnt or an !;!xploration - and then turn to the known fact.~ to prove it. Sometimes, an attempt to find proof refutes the new theory and it must be rejected .

Sometimes, finding a proof appears an intractahlt' problem and the theorem rt'mai ns unproven - for hundreds of years in somt' cascs. PROOF BY DEDUCTION

the Greeks out was refined and defined more rigorously much later is indirect proof. There are several types of indirect proof, including proof by contradiction and proof by reductio ad absurdulJI. Proof by contradiction aims to prove a statement is true by showing that itS opposite is nOt trut'. Proof by redllctio fld abSlmillni aims to prove a statement is true by using it to prove untrue something that is known to be true (so producing an absurd result). Hipassus' proof of the existence of irrational num hers was all indirect proof and is the earli est known.

Proof hy deduction works in small steps to deduce new truths from known truth~. For example, if we say. 'Humans are mammals' and ' Peter is a human' we can then say. PROOF BY INDUCTION 'Peter is a mammal' . Deduction is nl)[ The Greek model of proof was followed by wholly reliahle, even if the initial StatementS the Arab mathematicians and taken over are genuinely true, as the reasoning may nOt from them in the Middle Ages by early be valid. So we might say, 'Humans arc European scholars. But in [575 a new mammals' and 'Peter is a mammal', therefore 'Perer is :::t human' but the fir.~t All HORSES ARE THE SAME COLOUR The Hungarian mathematician George Pa[ya (1887-1985) statementS would al.~o he true if Peter were a dog or a used proof by induction to show that all horses are the hamstt'r or any other same colour. The case fo rn '" 1 (one horse) is dear - a horse mammal. Proof by deduction can only be the same colour as itself. Now assume the is nOt accepted as sufficiently theory is correct for n "" m horses. We have a set of m horses, all the sameco[ou r (1, 2, 3, ... m). There isa second rigormL~ by modern mathematicians, though it set of (m + 1) horses (1, 2, 3, .. . m + 1). We take out one was used extensively by the horse from this last set, so that it contains horses (2, 3, . Ancient Greek..; :mcl by m + 1). The two sets overlap; this second set is a set of m medieval mathematicians. horses, which we know is a set of horses the same colour. Parmenides is credited with By the principle of induction we can continue this for all the first proof by deduction further horses, therefore all horses are the same colour. in the 5th cenrury Be The argument is, of course, invalid as the statement is not true. The crucial point is that when n '" 2 the stat£'ment does INDIRECT PROOF not hold true: fo r this value, the sets do not overlap (the first Another meth od of proof contains only horse 1, the second contains only horse 2). which also originated with 198

PROB U MS ANO PR OOfS

DAVID HILBERT (1862- 1943)

David Hilbert is considered one of the most influential mathematicians of the 19th

He began as a pure mathematician and, when he turned his attention to

and 20th centuries. He was born

1912,

was

what

he

in East Prussi a in lin area that is now

part of Russia. As a

student, he met Hermann

considered

the

approach to

sloppy

math~

most

taken

physicists.

Minkowski and the two

by

stayed lifelong friends,

Hilbert also devised a

each

conceptual space that

mathematical

had infinite dimensions

cross-fertilizing other's ideas.

Hilbert worked in many

(called a Hilbert space). and

his

students

to the maths for his contributions to the

behind Einste.in's Theory of

axiomatization of mathematics.

Relativity and quantum mechanics.

modd emcrg~d; in Ar;lbmeticorlfnl Libri Dllo Francesco Maurolico (1494-15 75) b'llVC thc fi n;;t known description of mathematical induction, though him.~ of this method can be found earlier in worh by Bhaskara :md :ll-Karaji (C.AD lOOO). Proof by inductio n was also developed independen tly by Jako b Bernoulli, Blai se Pascal and Pi erre de Fermat. Proof by induction works by showin g firstly tim a hypothesis holds true for a first value (often n = I), then that it holds true for a later value (.~'ay n '" Ill) and also for the following value (n '" m + I). From the demonstration that it holds trw: for n '" III and n '" m + I it can be interred that this process could he repeated indefinitely to prove that it holds true for all further values. It's a bit like a row of dominoes, arranged on

end and equally spaced SO that if one falls It will knock the next over. If knocking th e fir.~ t do mino ovcr causcs til e ncxt to fall, it will in evit:lbly follow that they will all fall. A1aurolico used proof by induction to demonstrate that the sum of the first 11 odd imeb't'rs is n ~ : 1 + 3 + .5 + 7 + 9 + ... n -'" n

1

ASKING QUESTIONS

VVith the advcnt of calculus, complex numbers and later non-Euclidean bTL'Omctri es, more and morc W:lS denllUlded of proof. Berkel ey's o bjection to calcu lus as dealing with th" 'ghosts of quantitiL'S' was :I spur to greater rigour, nOt o nly in definin g the quantities and concepts with which mathematicians were working but in providing proofs. 199

P ROVIN G IT

The earliest writer Oil logic, in 1945, a boy in eighth grade taking part in a maths Pbru, died ill 347 or 348m:. Plato presents his Olympiad in Russia won first prize even though he did not philosophical works in the attempt to solve even one problem. The prize was foml of dialogues, o r awarded on the basis of a remark he submitted with an unfinished proof: con\'ers~tions, between ' I spent much time trying to prove that a straight line philosophers. They read as argum~nrs, with each can't intersect three sides of a triangle in their interior participant putting fiJn\'ard points but failed for, to my consternation, I realized that I his case in a series of have no notion 01 what a straight line is.' Statements which his opponent tht!l1. r~futes and Bur it '\~JS the 19th century which s:).w he thell defends. The argument uecomt::s the b'Teat revolution in mathematic:).l proof int:rea~ing:ly l"Olllple.x as the subject is as new methods of logic were developed and tackled rigorously. This m ethod, called people for the first time tried to apply dialectic, fonned the model for logical formal logic to mathematics. This requi.red debate until the ."'liddle Ages. Although a reassessment of the very basis of logic was a major conCl~m of these medieval mathematics and brought mathematics and scho lars, they did not think to apply it to philosophy rugNher. Mathematici:).ns, mathematics. It took more than 2,000 years unsettled Ly rL'Cem discoveries that threw for logic and mathematics to come together long-aL'Cepted truths into doubt, sought properly. new proof~ and questioned L"Vcn the most fundamental ideas underpinning their MATHEMATICS BECOMES LOGI CAL discipline. Suddenly, nothing could be taken One of the first ru tadde the issue was the for granted . Italian mathematician Giuseppe Peano (1858- 1932). H e wanted to develop the Being logical whole of mathematics from fundamental At the end of the 19th century and starr of propositions using formal log:ic . He the 20th century there was a flurry of developed a lo&,;c notation, butal~o a hybrid interest in the applieation of logi e to intern~tiollal lan6ruage which he hoped mathematics or, more precisely, the would be used for scho larship. Called derivation of mathematics frOm logic. It lnt~rlin~,'ua, it was based un the vocabulary came about largdy as a rcsult of rapid of Latin, French, German and English, but changes in mathematics and irs applications, used a very simple grammar. His use of it and critici sms of its rigour and \':llidity. hindered the acceptance of his Proof in mathematics is only parr of thc mathematical work. larger tOpic of logic which has developed The breakthrough III relating and grown since the time of the Ancient mathematic.~ to logi c came wlth the work of Greeks.

IGNORANCE EQUALS W ISDOM

'00

rigorolL~

B U NG LOG ICAL

the German logician and mathematician Gotdob Frege (1848- 1925), who has sometimes been called the gre:lt~t IObrici:ln since Aristotle. He set out to prove that all arithmetic could he derived logically from a set of hasic axioms and he is essentially the founder of mathematical logic. H i:' devised a way of represenring IObric using varia hIes and functions. A SEARCH FOR NEW AXIOMS

The German mathematician Davit! I·Elbert laid the foundations for the formalist movement that grew upin the 20th century by requiring that all mathematics should depend on fundamental axioms from whieh everything else ean be proven. He required any syStem to he both complete :lnd consistent, incapable of throwing up any contradictions fi-om the application of its axioms. He reformulated Euclid'.~ axioms himself as the fir.~t step in trying to find this faultless axiomatic basis for maths. Hilhert ramously proposed 23 problems which were still to be solved in 1900. Th~e effectively set Ollt the agenda for 20th century mathematicians. The mOSt important of Hilbert's problems for the dL'Velopment of logie in mathematics is the second . Hc proposes that it is neeessary to ser up a systcm of axio ms 'wh ich contains all exact and complete descrip tion' of the relations between basic ideas and requires ' that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory resultS'. In parricular, this was seen as a call for axioms to prove the basics of Peano arithmetic.

An~...vering Hilbert's call for an axiomatic basis for all mathematics. Bertrand Russell and Alfred North Whitehead published the three-volume PriIJfipia A1atbrmflfica in 1910- 13. Ambiriously named after NeWtOn's seminal work with the same title, the book aims to deriw all of mathematics from a sct of hasic a.xioms using the ."ymbolic logic set forth by Frege. It CO\'Crs only set theory, cardinal numbers, ordinal numbers and real numbers. A planned volume to cover breomt:'-try w"as abandoned as the authors were tired of the work. After getting a good way into the work. Russell discovered that a lor of the ~';J.me ground had been covered by Frcge and he added an appendix pointing Out the differences and acknowled bring Fregc's prior publiL';J.tion. The test of the Prif/cipil/ re.~ted on whether it was complete and consistent in H ilhert's tl'l"ms - L"1)uld a mathematica l statement he found that could not be proven or Jisproven by Prillcipil/'s methods, and cCluld any contradictions be produced using itS axioms?

MOVING THE GOALPOSTS

Before Prillcipia had a chance to stand the test of time, the key questions were taken away by German mathematician Kurt COdel. He produced twO 'incompleteness theorems' ( \ !J31) which dealt wi th Hilbert's proposal for the n, Seto 19l ••tro!.he ~9 , 10 ••trononl)' calculus in 16:> trigonometry Ul ~ 5 · 1i, ~ 7 _ 8 .. ion"tic..,t theory 191

ni."

•

Il.>hl"Se, Ch..rles 4 1· 5 Il.>hyloni:1n n",then",tics '" JI/".p.,,,,,,iu Il.>rl",rl" ... ,lox 190

206

n ·5

B.. yes'theo",m 17l·4 B.. yes, Thrn",. 17J· 4 Il«le n heU cu,\'< 179 Beltr::lm~ Eugenio I ll, 115 Berkeley, Bi.hopGeorge IS6 B"mouU~ D .. niel 171 B"mouU~J.l:nh 54, 157·8, 1 6 ~ B"mouU~Jolunn 157·8 B"mouU~ ~icol,", 171 B"mouUi n~ml",rs 4 5 B!.ck,,\hx In Bolpi,J'''''' I ll · 14 Bomhel~, Rof:oel 55, IH, 116 Bo.se, Ah...h"", 109 Bowl"y,Arthur I ~I a",lhu,)" K.y 165 Bourl .. k~ ~icob. 1~ 9 B... hn",gup" 1 7 · 1 ~ , 19, 87, Il~ Bn1bmuKYPtuiiMlklntu (B...hn"gup") 17· 18,

"

Brou"-er, L. E.J. 10l B runeUesch~ Fi~ppo

104- 5 butterflyeff

1" "'""',)'''''1'" I(~\.-I

(",,,,,,I. Hl ·' f"'g~,'.onJob lUI

1ri8",.om~t')'"'

fiuo.:trlog1C 191· '

,

I

I ~ I. 17')·1J)

j .cq .... rJ I"",,, +1

GC~>gr.Illhr (h~enll')

.t

107 ".n ,,". 70,71

g,,"',1ntti," • ..-,rrfric~1 4:', IIF. Ill. 151·1. Il6. 157 Lin,lpmonn. (::..1 L",," F""LrunJ lOll 9~ ~u,-'''' 1"'''1''' f-

1""loc

+-I ; 4 ,

LukU'"ic>.J'"

I~I-~

•

M•• i". $I.rip 116 .\t."i,"",. Ab ... han, ,I" 1"f'J

m"""r

~,

M""'yJlolJporo< 71

.\to.c~"f't".jobR )0. l"·",,1

"",u"eAm
;\;C)·u>ln.}'·"Y lal ;\;ighnn~'"" ,1""-",,~.", . Gi "'-. nn'

N · lO

Eh",hun 1\1)

01 ·,\1.> ....", IZ6

"u>p:ilUrymu"I",,, II

,\lLill",.,H.d

.... lucti,·e p"",f I 'Ill-Q infin.'), !" 1-j(,.8. 15(>-; in"I,"'" Z~ in«gr .."" 15O. 1 irTIIoolt>l num b..'1> II, ""'ll"I", ., lu!"-,, 14 7·11

G,Ule • •"UIle< 1~

n"ml ...""

207

INO£)(

inAociemlCg.pt IJ, I'l-lj.1i\l') in Andem C""",- If, m A",,'en, I.:ome I I, 19 ~I b-.. U .5 in rJuJU 14 chroJlQ!!,,"'" 21

",,'r!"-'

163

".l !i":H)

rr..",,,",-,

~nJ l lindu-ArobiC no!mf>e/" .... ,,,,,, 17·21 OJ 1ndi, 17 . ~nJint"~ 16

114

poly),,,,,," ""101",,, j!.J 1 '~o~j"'".I--,,",or

JlO"."

IW

4~ · 5 1

pml>ability 16~ ·74 I''''d .... 7J pm""",-" p:«~n 11>0· 1

",-.liJ I\!" of ,\Jc~ ... ,1ri;o

I'! ' ~

I