Contents
I. :->UMBERS AND fiNGERS
7
2. :'
17
3. NUMBERS AND THE ROMANS
25
4. NUM BERS AND ALI'HABETS
33
5. NUMBERS AND "NOTIII NG"
41
6.
NU~1BERS
INDEX
AND THE WORI.D
47 53
1. Numbers and fingers
M;ln)' thousands of yeai'S ago, when people asked the ques tion, " How many?.. 1hey nc:.-edt-'<1 to have nurnbei'S. Suppose you wanted to know how many sheep you had to make sure none h~ d b«n lost. Suppose you wanted to know how many days had passed s ince something had happened. Suppose you wanted to know ho'"' many s trangers were approaching carnp. People could show all they had ofsomething or mention each objcc1. If someone asked you how .nan)' days had passed sirlce the last time the tribe had killed a bear, you could say, "A day and another day and another day and an()thc:r day and another day." That's rather clumsy. h would be easy to lose track. Y<)u could make a oomparis()n with SOI'nt lhing else. You might notice thatjust near the river there was a group of trees. There was a tree and arlother tree and another tree and another tree and another tree. You could say, " The many days that have passed sir1ce 1he last time the lribc killed a bear is the same as the many trees that 7
are in that dump over there." That would really answer the q uest.ion. By looking at the trees, someone: could get an idea about how many days had passed. But will a perscm always be lucky enough to fin d a group or trees, flowers, rocks, or sta rs tha t are just as large a.s the group they are being a$ ked a bou1? \ Viii you always be able to point to some handy group and say, "That many .. ? I I \\'Ould be very nl« if you could arnnge to have groups of diCfercnt sit.c:s a lways nea_r you. Then whenever there was a " how many'' queuion, you could pick out the righ t group a nd .say, "That many''. Almost any person who though t tha t about how COil·· \'enient it would be: to ha\'Csuch groups would be likely to think of the fingen on one'1 hand$. Nothing (X)U)d be more C:Oil\'tnient than a person's own ha11ds. Look at your hands. £ach one has a finger and a nother finger and a nother finger a tld a nother finger and another fi nger and a nother finge r. You coul d hold up your hand and point to the 6ngcrs aJld say, "The many days that ha\'e pa.ssed since the laSl time the tribe killed a bear is the same as the many fingers I have on my hand. •• You can give a name to each finger. \\ 'ecalltheone that nicks out by itself the thumb. The ooe next to the thumb is thefimfmger, the ne.xl one is the middltfingtr, the next is the ring finger and fina lly there i$ the lilllt fin.gtr. You can hold up as many fingers as you want. You could hold up the forefinger and ben d alii he ooher finger> ou1 o(tht: way and s.ay, "This many''. Or ) 'OU could bold up a forefinger and a middle linger alld say, ''This man>•". You could hold up all the fingers on one hand and the ro1·cfinger on the other and say, "This many", a nd so on. It would be nice, however, if you didn' t iHtve to hold up
8
GIYing 1 n•m• to each finger
Middle finger Forefinger n r ~ index finger
9
your hands toshowoombinationso(fingc:rs. You migh1 be carrying something you didn' t want 10 pu1 do\\'n. It might be cold and you might not wam 10 ~xposc your fin.gen to th~ icy ";nd. It might be dark so 1ha1 no one couJd see )'Our fingers anyway. Suppose you made: up a word for each combination of fingers. For example, instead of holding up the forefinger only and saying, "TI1is many'', you could usc the word ~>One''. Then instead of having to hold up the forefinge r and Si:ly, " I have this many knives," you couJd say, " 1 have one knife." You could say that wilh your hands in your pocket o r in the dark and people would still under~ stand. \Vhy should you use the word one? Why not some other word? Nobody really kno\\-s. The word was made up so many 1housa.nds of years ago that we don' t know how it came about. h was used long bc(ort the modem Jan~ guages of Europe developed. Each modern t: uropean language u.5cs a d ifferent version of the word but they are all s imilar. In l:':nglish we say o,t, in French the "''ord is un, in Spanish it is UliO, it1German it is dn, in 1..1tin it is MtiUS in • Creek it i.s monos. All the words have the Jc:u er n in it. They all come from some originaJ word that we no longer know. We won't worry about the original word, however, or abou1 the \\-"Ord.s used in other languages. \\"e'IJ ju.st use the Engli.sh \\'Ords with which we are ramiliu. Forth<: combination of forefinger and middle finger, we s.ay two. Forefinger, middle finger and ring Rngcr i.s thrtt. Then we say rour, five~ six, scvtn, eight, nine and ten. The word ten is used when we hold up both hands with all the fingers l tretched out and say, "This is how many
simple to describe " how many... You can say, "I saw you six days ago," or "Bring in eig-h t logs for the fire," or ·~Give me two a.rrow.s." Then, suppose someone: drops a bunch of arrows at your feel and says, .. Hue are some arrows. I don't know how many there are." You can then count them. You can pick up an arrow and say, ''One". You can pick up another and say, ..Two". If when you pick up the last arrow you say, "Seven'', th:u means there are seven nrTows. Because you have ten fingers altogether, you have ten difTerenl words u$ed for ans\"tring qucstion5 of''how many?'' Such words are called m1mbtrs. But it is quite easy 1.0 have a group made up of more than cen things. Suppose )'OU had a bunch of arrows and picked them uponcbyone, countinsas you did so. Finally you pidccd up an arrow and said, ••Ten'', but found thert we.re: stiiJ some arrows on the ground. \Vhat do you do r\0\\'? You need more: numbers. J(you keep on making up new names for numbers, it is hard to remember them all. Ten different number§.-One, two, threc1 rour, five. six, seven, eight, nine, and tcn--.are quite enough to remember. But suppose you use those: numbers 10 make up new numbers in some sen.sible way. Then it would be ea~y to remember the new numbers. for instance, you might pick up ten arrows and find chat tht:rt: is stiU one left on 1hc ground. You can say, ..There arc one-left arrows.." Ash happens, the number word eleven is a fonn of a very old English word meaning ~>one left". In the same way, twelve is a rorm of a VCt)' old English
1 have:."
Afier that it is even easier. Thirteen is a slightly lwisted wny of writing ''three and ten", If you wrote .. threc::1en''
Once people get used to these words, it becomes very
10
word meaning " two left",
II
Two
One
Four
Three
•
Six
'
Seven
12
Nine
Eight
•
Ten
13
that would be quile close tO writing thirteen. Fourteen is even closer to "four and ten" and then there is ftfteen, sixteen, seventeen) eighteen and nineteen. Nineteen is .,nine and ten ... One more than that is ~
14
ever necessary to go beyond the word "thou!>and" • so we'll stop there.
'
2. Numbers and writing
Nobody knows when m.1mbcrs were fitst invented but certainly il was before ar1yone had invented writing. The time came. however, when people needed to work out a system for making marks that stood for words. This hap· pened about 5,000 ye-ars ago in the land we oow c-alllraq. Two rivers Row through 1ha1 land, the Euphrates and the Tigris. Ncar the place where they reach the sea 1here \•laS an anc,ient land we call Sumeria. lt was the Sumerians
who first made usc of writing. O ther people, the Chin~e and the Egyptians, also developed systems or writing. Gradually writing spread all over the wodd. \Vhen writing was inveoted, the Surnerians and the Egy ptians had dtics, temples, and f..""'.rmland irriga1ion d itc.hes. Many people had to cooper.Ht in the building of 1hes.e mark~ ofcivilisation. They all had to contribute time and eiTorc They also had to pa)' ta:<es. lt therefore bee-arne irnportam to keep records. The priests of the temples were in cha•·ge of such thirlgs. They had to make sure they knew who paid taxes and how
17
muC'h The' could memorise it, rxrhaps, but memory could pl3)• trich and thcrc:- might br argumenu. It was btuer to make some markings 1hat , .. ould show the state oftht' ta'll:t"$ m a ~rmancm way. In Cb('Ofargument the marlmg~ could be looked at. \\'hen writing "a5 fi rst im·ented the priests ustd a diffcr('nt marking for each word. This meant thtrc "ere a lot of markings w memorise, so leami1)g how lO read ar1d wri 1<· was \'Cry lwrd . In very ancient 1imts only the pricsu could rend ;uul w l'ite. Onr oft he most im portan t sets of ma1·ki n~s that had to be made was l(,r the diffcrcnc numlxn. Afie•· all. a ny r«
or
'"&*·
But sinrc the fi ngers "'ere used in connection with the inv('ntiOI) of number-words. whr not let the: numlx:•· one be •·cprclil'ntt.·cl b)' a s tra ight vcrtim l mnrk tha t loo ks like a f'in~t'r·? That is what the Egyptia ns did . for instance. They madf' .a mark like this, I, a nd that s tood for one. Any mark or symbol that is u~ 10 reprcscnt a number i~ called a 1111mnal. The stmbol I is an example of an Egyptian numeral. Othe-r people used the- sam(' srmbol or 310ntethm~ \eT) like it since e\'ti')'OOd) "ho thought ofon~ drew a picturt: of one finger. h isn't important what the exac1S)'mbol.s are, ho"e\'er. \\'hat is irnponant is how they a rc used. \ \'e can unders tand thi!J tx-u cr if we use symbols thtll nrr fam iliar to us. For the nurnbcl' one we can use the symbol I. Suppose now we wan t to write a symbol fo l' two. Ins tead ofirwcnting a brand new numeral, why not w rite I I? 18
)\ .. ..• ..
I
·,.• : -· ,,
..
.. •
19
This looks jUJI like two fingers. \ \'riling th~ next few numbers is. easy: I ll u three, I I II is four, 1J I l l is five, and so on, all the way up co I II I II ll I for nine.
Ancient EgypUans using the symbol T lor ten "'-! ~
II. '
--v:.
~~
\t r \t -"' \':!
""-
~ I;\
\t
~~-l ~~ ~"
The advan1ag-e: of this iJ that we know e:xac.tly what numbt-.r the syrnbol.s saand for by c::ouming the l 's. The
disadvantage is that when there a rc a considerable num· bcr of l 's it is wearisome co write them and tO count them. It is also easy to make a mistake in writing them or in counti1tg them.
~\t
,....~
¥
The Egyptiaru wua.lly "rotc the $)·mbo1J in some sort
ofpauem. for five they didn't write IIIII, 1hcywrote Ill and then wrote ll im med iately underneath. I t was easier
to see three marks and
IWO
marks than
10
make out five
mark.s ln a row. In the s.ame w~)' they wrote nine not as
111111111 but as th.-.., III's, one group under the other. Even dividing numerals into groups, howtvtr, will not heir, as we go into larger and larger numbers. lm agine writing fifl)•-rour as llll l l l lllllllllll lll l l lllll IIIII II lllllllllllllllllll . \\"hat the £g) ptians did ·was to ln,·em a new symbol for ten. for this they used a .srmbollike an upside-down U. We don't have to usc that syrnbol 1 however, to show how the Egyptian numerals worked. Suppose instead we usc the symbol T for ten. That \\OUid make it pa.n.jcularly clear to us bttau.sc. in our lansuage T is the first leuer of the word ten. If you wanted to write eleven you could write it 1'1 o r iT. It d OC"..$n't matter which one you write. h is tither ten and one or o ne and ten In either case the number is ciC'\·t:n~ You could have Til for twd,·e or liT or ew·n JTI. In any combination the symbol.s add up to twelve. [ t would be more convenient, howe,rer, to use some
..
•
•.:,. ..
'..... '
~.
.
, ._.,
t-•:...,..- t:1•
·:-
·~ '
' "'
'!
.'-
.,..
20 21
regular system. Then people would get used to it and be able to understand the numbers that much more easily. \Ve could put a11 the large numc:rals on the: left and all the small ones on the right. In lhat case twenty·thrct would be TTJ I I (ten and ten and one and one and one}. Sevent)'· four would be ITri"J"l I IIII and ninety-nine would be
You can go as high as you want in numbers by this method, inventing a new symbol every time you n« d ten of a lowt:r symbol.
J"ITI1"J"I"I I IIIIIIIII. Ofeourse the T's and l's would be arranged in patterns to make it easier to count thtm. The Egyptians decided that no more than nine of any symbol should bt written or counted and so the)' invented a new symbol every time ten of any symbol had to be written. To write one hundred you might write ten of the sym·
bois for ten like this: TlTliTITI"L Instead of doing that a new symbol was invented to stand for one hundred. The Egyptians used a kind of curl which looks a little like this: g. \Ve don,t need to usc that, howevc:r. Instead. let•s call tht symbol for one hundred H since that is the first letter of the word in our own language. T1uee hundred and thirty·thrce can be written HHHTITlll. Seven hundred and eighteen can be writ·
ten HHHHHHHTIIIIIIII and eight hundred and ninety can be wrillen HHHHHHHH I I I I I I I I I. You can write any number with these thrte symbols
up to nine hundred and ninety-nine, which is HHHHHHHHH ITITITITI IIIIIIIII. For a11 the numbers from one to nine hundred and ninety-nine you need to memorise. only three different symbols and you have to oount each symbol no higher Lhan nine. To writeooe thousand you would have to write ten symbols for one hundred, so a new symbol is invented. Another new symbol is invented for ten thousand, and stiiJ another for one hundred thousand. and so on.
22
23
3. Numbers and the Romans
The Egyptian4.!i)'Stem of numerals: gave special impor· tan cc to the number ten because that was the total num· ber of fingers on two hands. The Mayans, a people who lived in Southern Mexico in the days before the Europeans arrived there, used a sys· tem based on twenty. There arc twenly fingers and toes on your hands and feet. (In our own lang,u age we have some of that since we use Jcort to represent twemy. We can say that there are " two-score and roun een" people present, meaning ''fifiy-four''. ln the Gc::nysburg Address, President Lincoln began, •·f our-score and seven years ago,'' meaning "eighty-seven years ago".) However, it is also possible to give twelve special importance. The number twelve is more convcniem in some ways than ten. Ten carl. be divided only by 1wo and five. If you group things b y tens i1 isn'Lpossible to divide them evenly into thirds and quarters. Twelve, however, can be divided by Lwo, three, four and six. Some of 1he im portall.ce of twelve is shown by our use 2$
Atlantic Ocean
26
27
of•he word d4U1f. for example, we speak of a dozen eggs. Half a dozen is 1ix; a third of a dozen is four; a fourth of a dozen is three; a sixth of a dozen is two. \Ve go on to sell things in do2ens of douns. A dozen dozen is twelve twelves, which comes to one hundred and forty-four. \Ve call that a gro.s.s from a French word meaning large. The Sumerians gave s pecial importance to sixty. That can be: divided by still more numbers than can twelve. \r\'e also give importance tO sixty in our own lives. \Ve ha\'e sixty seconds in a minute. for example, and sixty minutes in an hour. The larger the number we base our system on, the more particular symbols we might have to count when we write the numbers. Suppose the Egyptians invented a new sym· bol only when they had 10 ust twelve some smaller symbol instead of1m. Then we might have toc:ountele\'en symbols insu:ad of only nine. Using twenty or six1y would be even worse. Suppose we used a number smaller than ten, though. We might usc Ave, for example, since this is the tota l number of Angers on one hand. About 2.000 years ago, large see1ions of Europe:., Asia, and Africa wert' ruled from thecityofRome.. This "Roman Empire" ustd a sy1tem or numeral$ based on five. The Romans ustd symbob taken from their alphabet. For· 1unately, the people of Europe and America use: the Roman alphabet so the Roman symbols are familiar tO us. The Romans began by letting one be written as I. For two, three and rour they had II, Ill, and Ill!. So rar it looks like lht Egyptian system but the Romans only allowed four of any symbol to be: used be(orc inven1ing a new symbol. lnstud ofwri1ing 6vc as the Egyptian Jill I they wrote it as V. Instead of writing six as 111111 they wrote it VI. Nine
or
28
was VIlli. If they wrote ten :u VIlli I, that would mean 6veoflhe symbol I and they didn't allow that. They used a new symbol for ten whieh was X. The li11 of symbols up 10 one tholl$and is as follows: I= one V = five X= teo L = fifty C • one hundred 0 = five hundred M = one thousand By using spccialsymbob for five, fifty and five hundred, the Romans never had tO use more Lhan four of any of the symbols for one, ten, or one hundred. To write twcnly-two they put XXII. Seventy-three is LXXIII. Four hundred and eighteen is CCCCXVlll. One thousand nine hundred and ninety-nine is
MDCCCCLXXXXVIIII. If you try to write one thousand nine hundred and ninety-nine by the Egyptian system, you would need one "thousa1'1d" symbol, and nine symbols each for "hull· . ht dr cd .. , "t en " , and H oneu . Th at would mean twenty-e•g symbols all together. In Roman numerals only sixteen symbols are needed. The Egyptian system uses only (our different kinds of symbols while the Roman system uses seven. In the Roman system you need less counting but more memorising. \Vhe:n these Roman numerals were first developed i1 didn'l matter in what order the symbols were placed. \Vhetheryou wrote XVI,or XIV, or- IXV,or VI X, it aU came 10 sixteen. No mauer what t.lw: order in which you add ten, five and one you end up with sixteen. Ofcourse, it is easier to add up a number if you arrange:
29
Roman symbols lor nineteen hundred and ninety-nine
30
31
the
I) mbob accordin~ 10
somr
con,~t
S) stern 1M
usuaJ 'ftot~ tJ to put all tM &) mbob of th~ Umt' iOf"t cogcthcr. The l;uxat symbol is on l~ ld't and ~5 )OU fl10\C co che right you "- ri1r down smaller and sma llrr symbol!. ThUJ .scverll)'•dght would always be wrilltn LXXVI ll, -.-orking down from L to X to \' to I.
4. Numbe r s and alphabe ts
Th< Romam lhm thou&f>t ol • "'•Y ol 11111 funhu pa.n•cular symbol that had to be wriuc:n do-.n. As long asS) mbols \Ole:~ al"'
d«~asi~ tlw numbtr ola
When you put thr smaller S)•mbol afttr 1hc la"Rrr one an tht usual way )OU add 1hr 1wo. TIM:rt:fol'(', \'1 is .. five plus one·· or sO. If on the other hand )OU putt~ Jmaller
S)mbol IN.fort tM LJorsoronc: )<>
CCCCLXXXX\'1111K'\~n symbols mstco~d of sixteen Of course, OllC~ )OU start u.s in~ 1hc sublracting notion, y-ou can't scramblt 1hc order of the 1ymbols a.n)' more. Jt brromes imponotnl to pia" uch s:ymboJ cx~Cd) lnr -.-utttn pari ol dw- Jlom,;an f.mpi..rt- brolr Ufl JUSt aboutl,500)<>n ago. Th< p
\\'c: have to l't'peat symbols in both the •:gyp11an and Roman S)'tlcm ol'numerals. \Vc always have 10 ha\'e com~ binationJ liLt Ill or XX or 1"1 I I I r. That means we have 10 count the symbols and that -.e m~ht make a mina.k~.
Is thc:u any way of nncr UJing any S) mbol more than once in any number? I f we want to do that we have to we a larger number ofsrmbols. 1f we don'1 warn two to be I I we have 10 gh t n a .sp«ial symbol. The same •ith three, ~with fOur. and so on It would S«m that 1h•• un•t so good sJ~ 'Wt' ~·ould have to do a lot or m~monJintC But suppoK" the S)'mbols were fl/r~' mtmorised. About 3,400 y.b
33
Phoenician alphabet and numbers
The alphabet spread in all direcLioos. h spread to the Hebrews and to the Greeks~ for instance. £veryone who learned how to wrilc (and this became much simpler with an alphabet) had to memorise the a lphabet. Ofcourse, the names of the lcncrs were diiTereot in different languages, but each group of people memorised the letters in ilS own lallguagc. Hebrew ch i ld ren ~ in learning the alphabet, learned to say: ahph, beth, gimmel, dated, !tay, r.·uv, and so on. Greek children learned to say: alpho, beta, gamma, delln, epsilon,
and so on . Childrerl who speak English leatrl to say: a;', bu, see, du, ee, ef, gee, and so on.
<.tltl, tla,
The alphabet is learned so weJI that it beC<>mes automatic for anyone who can wr1te. All the letters are knO\\•n in Lhe exacLordt:r and each one ha.<; a symbol. \ •Vh)• not let the symbols for the letters serve as S)•mbols for 1he numbers also? You can leLLhe firs LJeu er stand for lhe firs t number, the second letter fo r the second number, Lhe third letter for the third number, and so on. You don'L have to learn a single new symbol. You know all the S)'m~ bois already. The H ebrew letters and the Greek letters a re q uite diflCrcnt from our own but we don 1 t have to bother with those letters. We're just interested in the system of writ.ing numera ls ,.,..hic::h boLh the Greeks and Lhe Hebrews used. \\'e can just as well usc our own alphabet to see how i l works. U sing our own alphabet we c:;an say that: A =one
(
8 = two = lhree
C ..
D
~ four
E =five
35
Hebrew alphabet
Greek alphabet
F• six
c- ~~e'·en H~dght
,_, N ~
l
'T Tl '
<
~
0
';!
~
-, ;s
~
~ tj
~
:;
., w ]:1
~
; 36
I • nine J =ten
A N
--
B
r
A
t"lo" •cver, we can s tan combining. We can write eleven a.s ;'ten-one" o r JA. Twelve would be: " ten-t,,•o" orJ B. I n
0 TT
the same " 'ay we could go on wit.hJC for thineen,JD for fourceen,JE for fiftun,Jf for sixtccn,JG for ~-cntttn, J II for etghtttn, and J t for mn~t«n. \\'e could write: ..U for twenty, but now we \\'Ould be repeating symbols. Instead we go on to the next letter and let K sHmd for twen ty. In r.1 ct, starting withj , we could go on like this: J • ten K =twenty L • thiny
z z
H T e 1' •
'
K
alpha bet
•
E p
I
l rwc ltcepon going this wily, wc:' ll only reach the numbc:r twenty-six s ince there arc only twenty-six leu crs in the
)(
A 'f
Mn
M = fon) X • fifty 0 =sixty
P • seventy Q = eighty R • nillcty S =one hundred T •two hund~ {.; = 1h...., hundred
\' = rour hundred \\' = fiv< hundro:d X =six hundred y =seven hundred Z =eight hundred
37
\·V~'ve run out of1he ::tll)habet now buc we can rnake up a symbol jus. 10 carry things lO nine hundred. We can Itt that be "'riucn as £, for in.nance.. Sy this I)'S! em of numerals we can write any number under a 1housand wiLh one., two, or thrte S) mbols and in no number ,,.·iiJ any S) mbol be repeated. Se\-ent)•...fi\-c ls PE, one: huncb'ed and fitty-si~ is SNF, eight hundred and t\0.0 is ZB. nine hundred and ninety· nine is £RI. You can "'rite: all the numbers from one to nine hundred and ninet)'·njne yourself by this system. You will find it very ca.sy. Ifyou want to go beyond nirle hutldred and ninety· nine you can make special marks. A little bar over a letter mighl muhiply lht nurnlxrtha• repre.em~ by a Lhousand. Jn Lhis wny, A would be one lhousand 1 B wot1ld be two thousttnd, !lnd so on. five thousand eight hurldJ'ed and twent )'·One \~tOuld be f:ZKA. One bad poim nbout using Lhesarne symbols for letltl'S and numerals is that ic makes numbers look like words. For inslantt, using our own alphabet, the s-ymbol for five hundl'ed and siXt)·fivc is \VOE. That happens to lx a word 1hat mearu sorrow. for that reason, people might d~de aha a fi, t hundred and sixty-fh·e is a bad-luck number. They mir;hr create a "'hole system for deciding " 'hat numbers mean accordmg to tM leue.rs used as symbols. The Gruk and Hebrew people bolh di
38
Using the same symbols for leners and numerals can make numbers took like words
f
(
39
5. Numbers and
"nothing"
h might be beu er if we used a different symbol for ead ' of the numbers bUl didn't use Jcuers of the alphabet. S uppose we invented completely dilfcrcnt symbols. The H indu people or India developed a set or S)'mbols for numbers and we use them today. The forms we use
aren't quite the same as the ones the Hindus had many cemurics ago. Still if we look a t the old Hindu numera ls we can see lhc beginnings of the numerals we now use. From the Hindus we get the follo wing: I = one 2:::: l WO 3 =three 4 = four
5 =five 6;;; six 7 =seven 8 =eight 9 = nine
These numerals, or lhcir original anceslors, firs1 ap• 41
pearcd in India about 2,200 years ago. You mighc wonder how we came to use numbers like the Hindus. Arcn 1t they. after all,just another scl of sym~ bois? Wouldn't peopleratherstick 10 the old symboluuch as the Roman num~rals they were used co? Yes, people would! They did stick 10 the old symbol$ ju3t as long as they could . H owever, the rt3JOn why the Hindu numerals started spreading beyond India was that the Hindu pcQple had come up with a better idea. To begin with the Hindus (like I he Egyptians) made up new symbols for the numbers higher th an nine. They had d ifferent symbols for ten, twenty, thin y, and so Olli also for one hundred, two hundred, three hundred, and so on. Bu1 then JOmconc (and of course we don'l know who) must have asked why !his was ne<euary. The number two hundred means '"''-o ..one hundreds". The number twenty means cwo "tens". The number- two means lwo ..ones.,. In each cue the number means two of something. Suppose you make up your mind the symbol on the extreme right of a numeral tells you how many ones there a re:. The symbol to its left tells yot• how many tens there a re. The symbol next to the lefi tells you how rna.ny one hundreds there are, and so on. The meaning of a symbol depends on che position it is in. Jn thi.s way Lhe nine H indu numerals.-I, 2, 3, 4, 5, 6, 7, 8, s-.mighc be enough. Suppose )'OU ha\'e the numeral 3!'>4. The runhesl rigbl S)'mbols tells you tha.lthereare four ones or four. The one to iiS Jcfatcll.s you that there an five tens or fifiy. The one to irs leO tells you 1here are thr~ one hundrccb or three hundred. four and fifty and three hundred add up 10 three hundred and fifty·four and that is what 3M stands for. Any number can be read that way. 1ne number 18 s tands for one ten and eight one3, which comes to ten and eight, or eighreen. The number 999 s ta nds for nine one
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hundNxb, nine tens and nine ones. This Ls nine hundred and ninety and nine, or nine hundred and ninety-nine. You an go as high as you want in the Hindu system. A numbt:r like 87235 means., if you \\-ork from right to left, fh•e ones, three tens, 1wo one hundreds, seven one thousand• and eigh11en thousands. Add these up and you come out "'i.th eighty-stven thous.and two hundred and thirty-five. You arc still u.sin..g the nine Hindu numerals, and nothing dse. But there's a problem. S uppose you wanted to write the number two thousand and three. This is made up of two ..one thousands" and chree ' 'ones''. There are no hundreds ac all and no tens chhcr. Can )'OU write: it 23, standing for two "one thousands.. and lhrre "ona"? U you do this, how do rou know !he 2 stands for two ••one thousands..? Maybt: it stands fort""' "one hundreds" or for l\\·o "ems". Maybe you could leave a space to show that the ..one hundreds'' and the " tens" are absent. You could write 2 3. Then som eone would sec that the "one hundreds" and the uteus., are skipped so that the 2 must stand for two "one thousands", But how can a person be s ure that the empty space holds two empty columns? Maybe il only holds one. Maybe it holds lhre<:. No, leaving an empty space isn't enough. You must have some symbo) that stands for "no tens at all" or "no hundreds at aun. It was a ~ry difficuh thing, however, to get the notion chat such a symbol was needed. It took thousands ofyears aner human beings started using numbcrsymbols for anyone to think of writing a symboletuu stands for "'nothing",
I
Nine hundred and ninety-nine wriHen using Hindu numerals and Roman symbols
\ 'Ve don' t know exactly v.rho it \\•as who finally thoug ht of using such :;a symbol. We 1hink it was a Hindu. \\'e don' t know for s ure " 'hen it happened. ~.Ja y bc it wa$ about I ,300 years ago.
The symbol we now use fOr "nothing" is just a circle with nothing in iL It is 0. The Hindus called it SUI!J<'
rneanir1g "nothing". Lct•sscc how this " nothingu wo•·ks. If\,'Cwant to write twenty-three vte see that this means t\•.ro "tens" and three "ones .. and we write it 23. lfv..-e warn to write two hund red
You can work om fo r yourself why two thousand Lhree hundred is 2300 and why two thousand and three is 2003. For that maucr ten is one "ten" a r)d no ones a t all so il is wriuen 10. Using the nine Hindt• symbols I, 2, 3, 4, 5, 6, 71 8, 9, and th e S)' ll'lbol for " nolhing .. , 0, any number at all can be w riucn easily. The•·e is never any doubt what coh.rmn each symbol is in.
and three, we have two "one ht~ndrcd s" , no ttnl> a t all, and three "ones". We therefore write it 203. How a bout two thousand a nd thirty? That is two "one thom;ands",no one huodreds at aU, three " tens", a nd no or1es at (til. h is written 2030.
44
45
6. Numbers and the world
Th-e•·e is no doubt tha t the Hindu sys1e.:m of m •merals, ind t•ding the S)'mbol fo r ''m) l hing", is the best evtr invented. You can wri1e big numbers with o nly a fCw symbols
and you don•t have to memorise more than ten symbols a ltogether no maucr how large a number you have to wri te. No•· do you make; ,.,..ords out of llumbers and get
con fus<..-d. ]\'lost importanl of a ll, it turned o ut
be easier to do arithmetic with numerals written according to the Hindu tO
system than according to any other syStem ever invented . J n ancient times, only people who had studied marthemalics a long time could d ivide numbers t1sing either Roman numerals or G reek m •merals. Using the H indu systern, school children can learn to d ivide withot•t mu-ch trouble. If you think long d ivision is hard, just try doitlg it wit.h Roman numerals! Once people discovered how much easier i1 was to do a rithmetic with the Hindu mJmerals they used them also. The; Hindu syslem began lO spread . 47
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About 800 AO, not very long a fter the symbol fOr " noth· ing" was in.veJHed, the Hindu numerals spread into the lands north and \•l est of fndia . These lands were inhabited by people who spoke Arabic. Arabic·speaking people a lso lived all :;across nonhern Africa and in Spain as well . The Hindu numerals spread through Africa and into Spain. The Ara bs called the Hindu SUt!)'ll', the S)rmbol for " nothing", Jijr. About 820 AD, an Arabic mathematician named Muha mmed Al·Khwarizmi wrote a n important book on mathematics. He was the fi rs t to give fUll instructions on how to use the Hindu numerals in arithmetic. Over a hundred ye-ars later, a Frenchman na med Ge.-. bel't '''as very iruere.sted in gathering knowledge. At that time, France, England, and Germa ny were in a " Dark Age". There were few schools and books and hardly a nyone could read a nd write. Spain, ho'"ever, which was under the cont r()) of the Arabs, was much more advanced. Cerbert travelled to Spain in 967 ·" ' and stud ied books in Arabic. He came at ro.ss AI-Khwarizmi's book and was s truck with the convenience of the new system of numera ls. He brought them back to Fr:;ance v.:ith him. The people in Europe called thern A rnbit 11umeralt because they obtained lhem from A rab i c-s pe~ki ng people. The people in Eul'ope did not know that they carn e from Ind ia to begin with. We call the num bers I, 2, 3, a nd soon, Arabic numel'als toda)'· Since Cc.:rbert became Pope under the name of Sylvester II in 999 AU, you thiok European!; would li!itCn to him . l'hey didn't. A fe, ... other le-arned people ··ecommended the new Arabic m•merals. However, the Europeans of the time were using Romarl numera ls and were used to th em. Even though the Roman numerals were very clumsy aod made a rithmetic very difficult. the Europea ns s tuck with them.
49
T wo more centuries passed. Then we came LO Leonardo f ibonacci, a man who livc::d in an Italian city called Pisa. He pic:ked up the notion of the Hindu system of numerals while he was visiting northern Africa. In 1202, he published a book in which he u.s«! Arabic numerals plus the symbol fo r ''nothing". He showed how it could be used
Sl range symbols for their leuers and words you will find the familiar, I, 2, 3, 4, 5, 6, 1, 8, 9, and 0. And the whole thing staned when some prim itive person wondered how 10 describe how many stone axes he had a nd looked ao his fingers 10 sec if ohey would help.
in ~ri thm e tic. By that time, Europe had emerged from the .. Oark Age", People were more prosperous and more learned. In Italy, especially. there were many businessmen who had to do a lot of calcula ting to keep crack of their dealings. As I tali an businessmen found how convenient the Arabic numerals were they a ba ndoned the Roman numerals and used the new .syscem instead. They could see that the symbol for " nothing.. was most important. They used ohe Arabic word sift. Then ohey changed io 10 ~_1J1in since il was easier 10 pronounce and looked more natural. T he word zcpiro becom csz~ro to us a nd that i.! our most common word for the symbol 0 . O ne word !Omctimes a pplied to it 1 and a lmost as fami liar, is nougtu meaning ' 'nothing''. from llaly, Arabic numerals bega t'l to spread through lh< r<:so or Europe. By the oimc Columbus la nded in America all of Europe was us ing Arabie numerals. We usc Roman n umerals today, however, when we want to be impressive and when we don't need them to do arithmetic. Q ueen Elizabeth i.s the second English queen to have thai name. She is therefore called Elizabeth II. Po~ Paul was the sixth pope that name so he was known as Pope Pa ul VI. h is not only Europe, though , that use& Arable numerals now. In the last century, these numcf".als have spread every where. In dozens of strange languages using
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