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 .......... 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! "".ry<, parriro/Ilriy .\fi,hlltl A ,ui {ZiJIlO Jilltd (lIllrrllrd
FaCIIlrylCllmbridj(e FIlCIIlry/BMron lItA), Gordon Joly (London),
John Nllllj(hrvll (Camhridge Aillm '68, The Open Univ/'rriry
FIl,ulry),Jlldi SchofteM (London/GlIllrdilln Nru'!llnd .Iledill) Ilnd
Bill Tb01f'P50n (LonJonlCllmhridge Fllro/rylCiry UK Fllmlfy).

ftl'*
ARCTURUS

Arcturus Publishin g Limited

26/ 27 Bickels Yard
151-153 Bermondsey Su'eet
London SE 1 3HA

Published in association with

foulsham
w, Foulsham & Co , Ltd,
Th e Publi shing House, Benne tL.. Close, Cippen ham ,
Slough, Berkshire SLl 5AP, England

ISBN: 9i8-0-5i2-O:l41:1-9

This edition primed in 2008

Cop)Tight © 2008 Ar(!urus Puhlishing limited /Anne Rooney
(http://l-mw.annerooney.(o,uk)

All right~ rese rved

Covcr design and al't dirc(tion: Heatriz Waller

Design: Zoe Mellors

The Copyright A(! prohibits (sul~e( t to «:rtain \'Cry limited

excc ptions) the making of (opies of any ropyriglll \\'ork o r of a
su bstantial part ofsu(h a \\'ork, induding tllC making of (opies by
photo(opying or similar pro«:ss. Written pe rmission to make a (OpY
or (opics must therefore normaU)' be obtained from the publishe r in
fl(km(e, It is advisable also to (ons.uh the publishe r if in any doubt
as to the le~rality of any (opying which is to be ullt!t;rtaken.

British Library Cataloguing-in-Publi<:ation Data: a (ataloguc

re(ord for tllis book is available from tlH~ British LiIH'fII')'

I'rimed in China
 Contents
Introduction: The Magic of Numbers 6

Chapter 1 Starting with numbers 8

" ' here do numbers come from? • Numbers and bases • More numbers, big and small

Chapter 2 Numbers putto work 34

Putting two and twO tO~,'ether • Special numbers and sequences · Unspeakable numheN

Chapter 3 The shape ofthings 60

The measure of everything • Early breometry • Trigonometry

Chapter 4 In the round 92

Curves, circles and conics • Solid geometry • Seeing the world · Other worlds

Chapter 5 The magic formula 120

Algebra in the ancient world · The birth of albrebra • \Vriting equations
• Algebra comes into its own • The world is 1l(.'Vcr enough

Chapter 6 Grasping the infinite 144

Coming to terms with infinity . The emeq,rence of calculus • Calculus and beyond

Chapter 7 Numbers at work and play 166

Cheer up, it may never happen · Samples and statistics · Statistical mathematics

Chapter 8 The death of numbers 186

Set theory • Getting fuzzy

Cha pter 9 Provi ng it 194

Problems and proofs • Being logical • Mat were we talking about?

Glossary 204

Index 206
 I NTR O D UCTIO N

THE MAGIC OF NUMBERS

Think of 111111111herfrom 1 to 9. everything from th e hehaviour of sub-

Multiply it by 9. atomic particles to the expansion of the
If y ou have a two-digit 1lI1111ber, add the universe i~ Lased on mathematics.
digits together.
Take away 5. MATHS FROM THE START
Multiply the 111tl1lber by itself The earliest records of mathematical
activity - beyond counting - date from
The allswt!r is 16. How do es it- work ? 2,000 years ago. They come from the ferti le
It all depelUls 011 a crucial bit of 1l11mher ddtas of the Nile (Egypt) amI the pbins
'IIIagic: adding tog ether the digits of between the nVQ river:;:, the Tigris ~nd
1111tltiples of 9 always gives 9: Euphrates (Mesopotamia, now lraq). \Ve
9: 0+9=9 know little of the individual m:lthematicians
18: 1+8=9 of these l"arly l"Ulrures.
27: 2 + 7 = 9 a.lld so 011. Around 400sc the Ancient Greeks
developed an interest in mathematics. They
Th erellfter, it's all pltlin stliling: went heyond their predecessors in that they
9- 5 = 4,4 X 4=16. were interested in finding I"Ules that could
he applied to any problem of a similar type.
ThL')' worked on concepts in mathematics
here is plenty more magic in which und~rlie all th:!.t has come since.

T numbers. Long ago, some of Some of the greatest mathcmaticI:!.IlS of :!.Il

the earliest human ci~i l izations time lived in Greece and the Hell enic
discovered the strange and fascinating centre of Alexandria in Egypt.
quality of some numbers and wo\'t~ them A~ the Greek civi lization came ro an end,
into their superstitions and religions. mathematics in the \Vcst entered a dead
Numbers have entr:mced people ever since. zone. Several hundred years btcr, lslamic
and still hold the pOwer to unlock the scholars in th(· Nliddle East picked up the
universe for us, by providing a key to thr hOlton. Baghdad, built around 750, became a
secrets of science. Our understanding of dazzling intellectual centre where Arab

6
 I NT ftOD UcnON

Europe was struck by th e cataclysm of th e

Black Death (1347-50). Between a quarter
and a half of the populat ion died in many
European countrics. l t was the 16th cenOiry
before much new progress was made, bur
then there was a flurry of intellectual
acrh~ry, in mathematic~ as in science, art,
philosophy and music. The invention in
Europe of the printing prcss accelerated the
spread of new learning. European
Toledo ill Spaill htYonn rhe jfareilwy rbroujfh which mathematicians and scientistS began to
"Iff
Arah leamiT/jf fIIte11'd Europe ill rh,' J Irh cmmr)'- shape modern mathematics and to find
myriad applications for it.
i\-luslim scholars pulled tOgether the legacy \Vhile this has been the path of
of both Greek and Indian mathematicians development of present-day mathematics,
and forged something new and dynamic. many cultures have developed in parallel,
Their progress was b'l"eatly aided by their often making identical or comparable
adoption of the Hindu-Arabic number discoveries but nOt feeding into the main
system which we now use, and given .~tory centred on North Africa, the Middl e
impetus by their interests in astronomy and East and Europe. China kept itSelf separate
optics, as well as the requirementS of the from the rest of the world for thousands of
Islamic calendar and thl! need to find the years, and Chin(;!Se mathematics flouri shl!d
din'eoon of A-tecca. Howl-'Ver, the demands independently. Th e meso-American societies
of Islam wh ich were once a spur to in South America developed their own
dl!velopment eventually stifled further mathematical :.-yStems tOO, but they were
growth. Jv[uslim theol o~,'y ruled against wiped out by European invaders and
intellectual activity that was considered colonists who arrived in the 16th century.
spiritually dangerous - in that it might Early Ll(lian mathematics did feed into th e
uncover truths that should stay hidden, or Arab tl-adition from arowld the 9th centul1',
challenge the central mysteries of religion. and in recent years India has bt'comc a rich
Luckily, the Arab presence in Spain source of world-class mathematicians.
!lude the transfer of mathematical At the very end of our story, a single
knowlt'dgc to Europt' quite straightforward. number l>ystcm and mathematical ethos
From the late 11 th ct'nrury, Arab and Greek has sprL'ad around t he globL', and
texIS were translated into Latin and spread mathematicians from all culmrcs including
rapidly around Europe . Japan, [ndia, Russia and the US work
There was little new development in alongside those of Europe and thL' Middle
mathematics in Europe during the M:iddle East towards similar goals. Though
Ages. At the point where a few people were mathematics is now a global enterprise, it
equipped to carry mathematics forward, has only recently become so.

7
CHAPTER 1

STARTING
with numbers

Before we could have mathematics, we needed

numbers. Philosophers have argued for years
about the status of numbers, about whether
they have any real existence outside human
culture, just as they argue about whether
mathematics is invented or discovered. For
example, is there a sense in which the area
of a rectangle 'is' the multiple of two sides,
which is true independent of the activity of
mathematicians? Or is the whole a construct,
useful in making sense of the world as we
experience it, but not ' true' in any wider sense?
The German mathematician Leopold
Kronecker (1823-91) made many enemies when
he wrote, 'God made the integers; all else is
the work of man.' \Vhichever op.inion we incline
towards individually, it is with the positive
integers - the whole numbers above zero - that
humankind's mathematical journey began.

III tb~ b~gillllillg .. CIW~II/m C01i1d pllillt, bllt colild tbey COli lit?
~ ....n", wn" ""M""

CAN AN IMALS COUNT?

Could the mammoths count thei r
attackers? Some animals can apparently
count small numbers. Pigeons, magpies,
rats and monkeys have all been shown to
be able to count small quantities and
distinguish approximately between larger
quantities. Many animals can recognize if
one of their young is missin ~ too.

FOUR M AM MOTH S OR
M O RE MA MMOTHS?
Imagine an early human looking at a herd of
potcntiallullch - buffalo, perhaps, or woolly
mammoths. There arc a lot; the hunter has
no number system and can't count them. He
or she has a sense of whether it is a laq,rc
IV/" ngll"lte fill a~cts of0111' lift by III/mbns. bllt (bat herd or a small herd, recognizt!S that a
bas I/Of ak,ays hem rbe crISe. Tbe w;'lIIfe halld <1}flS
added ro clixJ:s ill 1-1-7'), tbe Sl'Colld halld arOlllld 1560.

Where do numbers
come from?
Numbers are so much a part of our
everyday lives that we take them for
granted. They're probably the first thing
you see in the mor ning as you glance at the
clock, and we all face a barrage of numbcrs
throughout the day. But there was a rime
before number systems and counting. The
discovery - or invention - of numbers was
one of the crucial stcps in the cultural and
civil development of humankind. It enahled
ownership, trade, science and art, as well as
the dL'vclopmellt of social Structures and
hierarchies - and, of course, brames, puzzles, Mlllly agaillst ollr is ilion likdy ro mSiIIl' a safr
sports, gambling, insurance and even OlltrOlllf alld a mM! for blllltrrs t''lllippt'd ollly
birthday parries! with prilllitive WMpollS.

10
 W HER£ 00 NUMB ERS COME FROM?

HOW TO COUNT SHEEP WITHOUT

COUNTING
As each sheep leaves the pen, make a notch
on a bone or put a pebble in a pile.
When it's sheep bedtime, cherk a notch or
a pebble for each sheep that comes in.
• If there are pebbles or notches
unaccounted for, go and look for the
lost sheep.
• If a sheep dies, lose a pebble or scratch
out a notch.
• If a sheep gives birth, add a pebble or
a notch.

single mammoth makes easier prey, and It isn't nec~sary to count to know
knows that if there arc morc hunters the whether ~ set of objects is complete.
task of hunting is hath easier and safer. \Ve c~n glance at a tahle with 100 places
There is a clear difference between one and set and see instantly whether there ~re
'more-than-one', and between many and any places without diners. One-to-one
few. But this is not counting. correspondence I S learned early by
At some point, it becom~ useful to children, who play games matching pegs to
quantify thc extra mammoths in some way - holes, toy Dears to beds, and so on, and was
or the extra people needed to hunt them. learned earlr br humankind. This is the
Precise numbers are still not absolutely basis of set tht..'Ory - th~t one group of
essential, unless the hunters want to objects can be compared with anothcr. We
compare their prowess. can deal simply with sets like this without a
concept of number. So the early farmer can
TAllY-HO! move pebbles from o ne pile to another
Moving on, and the mammoth hwlters without counting them.
settle to herdin g their own animals. As soon The Ilecd to record numbers of objects
as people star ted to keep animals, they led to thc first mark-m~king, the precursor
needed a way to keep track of them, to of writing. A wolf hone found in the
cht..'Ck whether all the sheep/goatslyaks/pigs Czech Repub lic carved with notches
were safely in the pen. The easiest way to do more than 30,000 years ago apparently
this is to match each animal to a mark or a rcprt..'Senrs a tally and is the oldest known
stone, using a tal/y. mathematical object.

11
 STARTIN G WITH NUMBlRS

FROM TWO TO
TWO·NESS ONE, TWO, A LOT
A tally stick (or pile of A tribe in Brazil, the Piraha, have words for only 'one', 'two'
pebbles) that h as been and 'many'. Scientists have found that not having words
developed for counting for numbers limits the tribe's concept of numbers. In an
sheep can bi.' pur to other experiment, they discovered that the Pirah;i could copy
uscs. If there arc thirty patterns of one, two or three objects, but made mistakes
sheep-rokens, they can also when asked to deal with four or more objects. Some
be used for tallying thirty philosophers consider it the strongest evidence yet fo r
gOatS or thirty fish or linguistic determinism - the theory that understanding is
thirty days. It's likely that ring.fenced by language and that, in some areas at least,
tallies were used early on to we can't think about things we don't have words for.
count time - moons or days
until the birth of a baby, for
example, or from planting to cropping. The concrete objects counted heralds a concept
realization that 'thirty' is a transferable idea of numher. Besides seeing: that four apples
and has some kind of independence of the can be shared out as two apples for each
of two people, pL'ople discovered that
four of anything can always be divided
into two b'TOUPS of two and, indeed, four
'is' twO twos.
Ar this point, counting became mort:
than mllying: and numbers nl.:!eded names.

BODY COUNTING
Many cultures developed methods of
counting: by using parts of the body. They
indicated different numbers by pointing at
body parts or distances on the body
following an established sequence .
Eventually, th!;' names of the body p:lrts
probably came to stand for the numbers and
'from nose to big toe' would mean (say) 34.
The body part could be used to d!;'note 34
sheep, or 34 trees, or 34 of allY thing else.

TOWARDS A NUMBER SYSTEM

How nllllly hln~ 71'e gotl A Porrugllese villryrmi Makin g a single mark for a .single counted
work" lIotcbes II Ulllystick ro "ecord Mcb bnrkt1 of object on a stick, slate or cave wall is all very
grllpn rhllr passer by. well for a small number of objects, but it

12
 WHER£ DO NUMBERS {OM£ FROM?

quickly becomes unmanagl.!able. BeJore In l\Jlesopotamia (current-day [raq), a

humankind ("(mid use numbers in any more simibr system existed from at least 3000Be.
complex way than simply tallying or A still-familiar simple grouping syStem
counting, we needed methods of recording is Roman numerals. Numbers 1 to 4 are
them that were easier to apprehend at a represented by vertica l str okes :
glance than a row of strokes or dotS. \Vhile
we tan only surmise from observing non- I, II, III, 1111
industrialized people as to how verba l
counting systems may have developed, The Romans gave up at Ill, switching
there is physical evidence in the form of to a symbol for five, V. Later, they
artefactS and records for tht:' development of sometimes used rv for illl. [n this case the
written number ~yswms. position of tbe vertit':!l stroke determines its
The earliest number systems were meaning - five minus one. In the same way,
related to tallies in that they began with a lX is used for nine (ten minus one).
series of marks corresponding one-tn-one Different symbols are used to denote
to counted objects, so 'lIT' or ' .. .' might multiples of five and ten:
represent 3. By HOOne, the Ancient
V 5 L 50 D 500
Egyptians had developed a system of
X", 10 C:: 100 M '" 1,000
symbols (or hieroglyphs) for powers of ten,
so that they used a stroke for each unit
and a symhol for 10, then a different symbol Numbers are llUilt up hy grouping unilS,
for 100, another for 1,000 and so on up to tens and so on. So 2008 is represented by
1,000,000. \Vithin t':!ch group, the symbol MMVlll. The characters for 5, 50 and 500
was repeated up to nine times, grouped in can't be lL~ed more than once in a number,
a consistent pattern to make the number since VV is represented by X, anrl so on.
easy to recognize. Some numbers are quite laborious
to write. For example. 38
is written XXA'VU]. The

~:~
system doesn't allow

1111' I!! '

subtraction from :Inything
I, II, III, II· 1111, III' 1111111, except the next symbol in the
numeric:!] sequence, so 4Y

I u ,. ~
can't be written IL (50 minus
I); it has to be written XLLX
(50 minus 10; \0 minus 1).
The next Step is a system
1,000 10,00() 100,000 1,000,000
which instead of repeating
the :.ymbols for a number
Em'~Y Egl'ptiml hhroglypbs repn!Si'lIIt d IlIIlIIbl'rs I~illg POW"" of tw, (A..,"\.,"'( for 30, for instance)
(lml cOllid sb{J'J) JIIlmben lip to 9.999,999. uses a ~ymbo l for each of the

"
'{.sl ",n,", W'ffi "'M" "

digits 1 to 9, and thell this is used with the shown by three digits. Roman llullu;,rals, on
symbols for 10, 100 and so un to show how the other hand, need between ant' :lIld four
many lOs, IOOs and 1,000s arc intended. digits for the numbers 1 to 10 and hetwc(;!11
Th e current Chinese system \rnrks on this one and eight digits for numbers up to 100.
principle . So:
CIPHERED SYSTEMS
11]-r- 4 x 10",40 The hicroglnJhic ..,ystem described above
(see page 13) was only one ofrh ree systems
but;-G: 10+4",14 uscd in Ancient Egypt. There were twO
cip hered systems, demotic and hi erati c. A
andlZll-rlZ!l 4 X 10+4 = 44 ciphered system nOt on ly has different
symho ls for the numerals I to 9, but
This is kn()wn as a multiplicative grouping distinct symbols for each of the. multiples of
system. The number of characters needed 10, 100 anti 1,000. H.ieratic is th e old est
to represent numbers is more regular with known ciphered system . It could e..'\: pre.~s
this typl! of ~ys [em. Numbers 1 to 10 are numhers in a very eompact form, hut ro use
shown by one digit; numbers 11 to 20 are it people mU St learn a large number of
shown by twO digit~; thereaher, multiples of different symbols. This may have served a
10 up to 90 :lrc shown by two digits (:20, 30 soeia J purpose, keeping numbers 'specia l'
ctc.) :md the orn er numbers up to 99 are and so endowing those wl1l) knew them

HOW OLD 15 THE COW AND HAVE YOU BEEN PAID?

In Babylon (from southern Iraq to the Persi an Gulf) two systems of writing numbers were
used. One, cuneiform, consists of wedge.shaped marks made by a stylus in damp clay
which was then baked. A different system, curvilinear, was made using the other end of
the stylus, which was round. The two scripts were used to represent numbers for different
purposes. Cuneiform was used to show the number of the year, the age of an animal and
wages dUE. Curvilinear was used to show wages that had already been paid.

(SO . 1) (60) 40 · 2 ~

(60)] .. 11 (60)'+ 47(60)+111 _ 2S11.~

 W HER { DO NUMB ER S COME FRO M ?

, , , ,
UN ITS
U 11\ ll.<j "\ '" - t. =? rt
~
-
~

TENS
A A 1\ >r 7J .::.I- 51 llll
HUNDREDS
~ )l ? ? ?," /3~.3
~ ! "i ~ ~ !!l; ~ ~ ~
THOUSANDS

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

with extra power, forming a mathematical position of the numerals to show their
elite. In many cultures, numbers have been meaning. This ean only work when there is
closely allied with divinity and magic, a symbol for zero, as otherwise there is no
and preserving the mystery of numbers way of distinguishing between num bers
helped to maint:lin the authority of the such as 14, 204 and 240, a problem
priesthood. Even the Catholic Church was encountered by the Babylonians.
to indulge in this
10,000 1,000 100 10 1
jealous b'l.lardianship
54,321 == 5 X 10,000 4 X 1,000 3 X 100 2 X 10 1 Xl
of numbers in the
European .M.iddle 10,070 == 1 >< 10,000 a x 1,000 a x 100 7 X 10 1 XO

Ages. Other cil>hcrcd

systems include Coptic, Hindu Brahmin, A positional system loan show very
Hebrew, Syrian and early Arabic. Ciphered large numbers as it does nOt need new
systems often use letters of the alphabet names or symbols each time a new power of
to represent numerals . lOis reached .
l1lt~ Mrli~t positional ~y;rem that em he
GETTING INTO POSITION dated was developed by the Sumerians from
Positional number syStems, such as au r own 3000 to 1OO0BC, but it was a complicated
modern SYStem, depend on the position of a system that used both 10 and 60 as its bases.
digit to sho\v its meaning. A positional It had no zero until the 3rd century Br:,
system de\'elops from a multiplicative leading to ambiguity and probably confusion.
grouping system such as Chinese by Even after zero was introduced, it was never
omitting the characters that represent 10, used at the end of numbers, SO it was only
100 and SO on and depending only on the possible to distinguish between, say, 2 and

IS
 STAItTlN G WITH NU MBllt~

SUMERIANS AND BABYLONIANS

The fertile area of Me~opotamia, between
the two river ~ ngri~ 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 the fourth
millennium BC had established 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 archai c letters rhey no longer used for
easy and sometimes not. The statement writing. For numbers over 999 they added a
'1 have 7 sons' was unlikely to be interpreted tick mark to the right of a letter to show rhat
as '1 have 70 SOilS' - but a statement such as it must be multiplied by a factor of 1,000
'An army of 3 is approaching' contains (like our comma as a separator) or the letter
dangerous ambiguity. An army of 300? No 11111 as a subscript to show multiplication by

problem. An army of 3,000, or 30,000 or 10,000. To distinguish numbcrs from

even 300,000 is a very different Illa tter. words, they drew a bar over numbers.
One of the two number systems in use in GrL'ek philosophers larer came up with
Ancient Greece, that most popular in methods of writing very large numbers, nor
Athens, used letters of the Greek alphabet because they especially needed them, but to
to represent numbers, bCb>lnning with alpha counter claims rhat larger numbcrs could
for I, DNIl tor 1 and SO on up ro 9. Next, nOt exist since t here waS no way of
individual letters were used for multiples of representing them.
ten and then for muh:iples of lOO, so that The Mayans used a complete positional
any three-digit number could be system, with a zero, used thoroughly. The
represented by three letters, any four-dib>lt earliest known use of zero in a Mayan
numher by four letters, and so on. They inscription is 36nc- Mayan culture was
didn't haw enough letters in their alphabet discovered - and consequently wiped out,
to make it up to 900 with this system, so along with the Mayan civilisation - by
some of the numerals were reprL'Sentcd by Spanish invaders who came to Yucatan in

16
 W HER£ DO NUMBUS COM E nOM ?

the early 16th century. The Mayan number M~()p otami:l

ahout ;\0650 refers to nin e
system was based on 5 and 20 rather than Hindu numbers.
10, and again had limitations. The first
perfect positiona l system was the work of 2 3 4 5 6 7 B 9
the Hindu s, who used a dot to represent a
= - + 1 ..., I
vacant position . " "
THE BIRTH OF Adding a diagonal line hetween the
H I NDU-ARABIC NUMBERS horizontal strokes of the Brahmi '2' and
The numhers we use today in the \Vest have a verti ca l lin e to the right of the strokes of
a long histOry and originated with the In dus the Brahmi '3' m:lkes recognizable versions
valley civilizations more than 2,000 years of our numeral~.
ago. They are first found in early Buddhist The Brahmi numerals were part of a
inseri ption s. ciphered loystcm, with separate ~ymb()ls for
The use of a single stroke t() stand tor 10,20,30 and SO 011.
'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
and t) to the li st, and 3 and 5 .\'. _h:i:..)4t1(+'?
are found in the N asik eaves 2"
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 mathematician al-Khwarizmi (c. 815),

Opening of tbr U7Iiverse) written by the and 011 the Use of tbr Indian Numerals by
IndiOln mathematician Brahmagupta in 618. the Arab Abu Yusuf Yaqllb ibn [shaq
The caliph had founded the 1-1ouse of al-Kindi (830).
Wisdom, an edueJtional institute that led A system of counting angles was adopted
intellectual development in the Middle East for depicting the numerals 1 to 9. It's easy
at the time, translating Hindi and Classical to see how the Hindu numerab could he
Greek texts into Arabic. H ere, the converted by the addition of joining lincs
BmlmltlsplJllftlsiddbtlllftl was translated into to fit this system - try counting the angles in
Arabic and Hindu numbers tOok their first the straight-lin e forms of the numerals
step tOwards the \;Vcst. we use nOw:
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
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
Europe not just the numerals and arithmetic methocls but
--1"1
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
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
Zero was adopted around the same time; its own symbol. But it did need a symbol
zero, of course, has no angles. The Arab when positional number systems emerged.
scholars devised th e full positional system Initially, a space or a dot was used to
we lISC now, abandoning th e ciphers for indicate that no figu re occupied a place;
multipl es of ten used by the Indian the earliest preserved use of this is from the
math ematicians. mid·2nd millennium Be in Babylon.
Not long after, the new fu sion o f The Mayans had a zero, represented by
Hindu-Arabi c number systems made il5 the shell glyph:

~
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 This was used from at least 368e, but
produced in Spain in 97 6. had no influence on mathematics in the
Old World. It may be that Meso-Americans
ROMA NS OUT! were the first people to use a form of zero.
Of course, Europe was already using a Zero Glme to the modern world from
number system when the Hin du-Arabi c India. The oldest known t ext to use zero is
nOtation arrived in j\'loori sh Spain. Mter the Jain Lokavibhaaga, dated AD458.
the fall of the Roman Empire in th e \Vest, Brahmagupta wrote rules for working
tradition ally dated A04 76, Roman culture with zero in arith metic in his
was only slowly eroded. Brahmasphutasiddhanta, setting out, for
Th e Roman num ber system was instance, that a number multiplied by zero
un chall cnbTCd for over 500 years. Alth ough gives zero. This is the earliest known text
th e Hindu -Arabi c numerals crop up in to treat zero as a number in its own right.
,\ fLow works produced or copied in th e AI·Khwarizmi introduced zero to the
10th century, they did not enter th e Arab world. The modern name, 'zero',
main stream for a long time. comes from the Arab word zephirum by
way of Venetian (the language spoken in
1 I Venice, Italy). The Venetian mathematician
5,000 (I)
5 V luca Pacioli ( 1 445~1514 or 1517)
10 X 10,000 (I) produced the first European text to use
50 L zero properly.
50,000 (I) While historians do not count a 'year
100 C
zero' between the years 1 Be and ADT,
500 0
1,000 M 100, 000 (I> 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 FibOlltlcci, the It,iliall

sophistication, the Romans needed larbrer matbeTlTaticiall, /efll1led
and larger numbers. They developed a abollt Hilldll-Al"flmc
system of enclosing figures in a box, or I/lfTlTemir ar a boy "/:.rbile
three sides of a box, to show that they tl"lwellillg ill .Nonb Afiiro
should be multiplied by 1,000 or 100,000. -;;;itl1 hir rrader/a/ber.
The system wasn't used consistently,
though, so Hindu-Arabic system,
particularly amongst
fV1 could me:ln either 5,000 or 500,000. tbe mercbants and
accountants. Even so,
Arithmetic is virtua lly impossible with it took many centuriC!:>
R oman numerals and this was to lead to its and considerable
eventual replacement. struggle before
Europe moved
XXXVIII + over completely to the use of the
XIX Hindu -Arabic system (see Unspeakable
LVI! (38 + 19 = 57) numbers, pa~,'c 56).
Roman numerals continued to he used
For the purposes of accounting, taxation, for many things long after they were
census taking and so on, Roman replaced in mathematit~Jl functions. They
account;lIlL~ always used <In abacus. Hindu-
Arabic numerals offered a considerable
advantage in that the positional system 'The nine Indian figures are:
made arithmetic w ith wrinen numbers very 987654321
easy. Botb Fibonacci (Leonardo Pisano, With these nine figures, and with the sign
[.1170-1250) and Luca Pacioli, both o .. any number may be written. '
better known for otber achievements, were Fibonacci, Liber Abaci, 1202
instrumental III popularizing the

20
 W HER£ 00 NUMBERS (OM£ FROM?

NOT OVER YET

CHRONOGRAMS It would be a mistake to
Phrases that incorporate a number in Roman numerals- think that our numbers have
chronograms -were often used on tombstones and books. stopped evolving. tn the last
By picking out certain letters and rearranging them, a date century we have seen the
is revealed. For example, My Day Is Closed In Immortality is development and :.ubscquent
a chronogram commemorating the death of Queen declin e of the zero with a
Elizabeth I of England in 1603. The capitals read MOO!! slash through it, 0, to
when put together, which corresponds to 1603. A coin distinguish it from capital
struck by Gustavus Adolphus in 1627 includes the latin ' 0' in computer printouts,
inscription Ch,lstVs DuX ergo tflVMphVs ('Christ the leader, and the reprl!Senration of
therefore triumphant') which is a chrono gram for digits as a collection of
MDCXWVII o r 1627. straight lines so that they can
be shown by illuminating
bars on an LED display.
are still often used on elock faCl!S, fur Computer-readable character sets, too, have
example, and to show the copyright date of been developed for usc on cheques and
movil!S and some TV programmes. other financial documenL~, taking our
numerals fur from their cursive origins.
In addition, we have developed new
types of notation for writing numbers
so unimaginably large that our anCl!Stors
could have had no conceivable usc for them
(sec pab'es 26- 33).

Lllm Pacioli 7L"IlS f1 Frallciscall frial: III rbis pmTrair Bar rodes lISe lilies I)f differe1l1 tbickllesses (I)

by Jm:I)/IO de Barb"'7 (I;. 1495), be if dmTl)llstratillg /"!'p/"~mt /llimbers: tbeY( all' reml by CIJmplltfl7zrd
I)/I~ I)f Eudid's rbel)/"ems. sca mnrs 'Il·bicb ~ce' tbem ar lilimbers.

21
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