Number Systems (Popular Lectures in Mathematics)

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,.Ntl .. be,r Systems. Popular Lectures _in Mathe,matics, • :hic!lgo 'tJniv •• Ill. Oept. of Mathematics. , NatiJnal Scie~ce Poundation,. Washington, D.C.

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ABSTRAcr

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The origin, p~opertias, and applications of va~i,~s numbe.; systess are diScussed. lmong the 15 tqpicsdiscuss,ed 'are: -tests for divis'±bility, the binart ,.sys~em,,, the galle of Nill" ' ,~omputer~, and inf111ite numbe,r re?'r~ser..tati'ons. (KK), .1, "

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Survey oI$cent Eiut European Mathemati

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A project conducted by Izaak Wirszup, Department of Mathematics, the University of Chicago.

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". :.:;,;',': ": ' ~';:'::i , 2. The Origin# of the Decimal Number .system, ' . . . ' , 3. Oth~r Number Systems and Their'.ptigins 4. Positional and Nonpositional Sy~ms 5., Arythmetjc Operations in Various;~umber Sys,tems' 6. Translating Numbers from One System 10 Al10ther 7. Tests for Divisibility ~,~~;t~ , 8. The Binary 'System .',. ' ':'" ~~' 9. The Game ofNim "':';,.' ,. , 10. The Binary"'Code and Tetegraphy 11. The Binary System-A Guar<Sian. of Secrets . It. A Few Words about Computers 1~ Why£lectronic Machines .. Prefer" the Binary System 14. One Remarkable Property of the Ternary System 15. On Infinite Nurpber Repr,esentations

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The language of numbers, like anY,language, has its own alphabet. • In the language of numbers 'that is !?OW used virtually worldwid~t the alphabet CGnsistsoften digits. 0 through 9. This langua~ is the decimal -number system. But-this languase has not always.been.(ssed universoYl~. From a pu~ly mathematical point of view. the decimal sy.stem has no inherentb.dvantages over other possible number systems; its popularity is due . not' to mathematical principle, but to a Set of historical and biological factors. . ~ . . \. . In ~nt times the decimal system has received serious competition fro~ the bjnary and ternary. systems, which are '! preferred" Qy modern computers. . ) " In this pamphlet \Ve will discuss the origin. properties, and applications of various number ~~tems ..The reader need not be familiar with mathematics beyond that covered in the high scpool curriculum .. Two'sections (9 and 11) have been added tp the second edition, and several minor corrections have been made.

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1. Rouod \~ UID'OUDded

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:' A man about 49 years ~ld went out for a ~on. MlaIked down the street for about 196 meters, and went into a store; he bought 2 dozen eggs'1he're and then continued walking.... " Doesn't that sound alittie strange rWhen we measure' something approximately, such as distance 'Or someone'sage, we always use round numbers and 'Ordinarily say . ( .. 200 meters," .. 50-year-.old man," and t~Jike. It 'is simpler to operate with .round numbers: They are easier to remember, 2nd arithmetical • computations are easier t~ perform on them. For.example,po 'One has· trouble multiplying 100 by 200 in- his head, but multiplying t\\'o un- . reu~three:digit numbers, such as 147 and 343, is so difficult that almos no one can ~o it without pencil and, paper. . ( I king of round numbers, we do not normally realize that the divisio~of numbers {nto .. r'Ound" and" unrounded" is .dependent on . the wat in which we are writing the nu~ber or, as we ~~uaUy·say. which number s,.}'stem we are using. In investigating this matter, let us first r exkmine1he' d'&imal number system, wnich we all use. In this system .... every positive integer' (whole number) is represented in the form of a . sum ~f ones, tens, hundreds, ~d so 'On; that is, in the form 'Of a,sum of " powers of 10 with coefficients which can assume the values 0 through 9. For example, the notation 2548 den~tes the number consisting of 8. ones, 4 tens,S hundreds, and 2 thousands; so that 2548 ifap abbreviation of the expression \

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Here every square contains product' of the numbers of the row column on whose intersection the square lies, with an numbers written in base six (we havc4 0mitted the subscripts in order to make the table more compact). . ', Using this table, it is easy to multiply by colum-ns numbers coa),aining

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Dividing" diagonally" is also possible in any number system. Consider a prohl/m like the following: ' Divide (120101)3 by (102)a . The solution is ".' ...:~;-~~ ,:-

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Since ~ I cannot be the base of a number system, x = 6, Thus, th~ teacher's answer was in the base six system, aod he had 36 pupils 16 boys and 20 girls. 6. Translating Numbe~ from One System to Another \
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On lrifinitcNumbcr Representations , "

further divide each of the two remaining parts into three equal parts and reject the center segments. After this there remain four small pieces, from each of which we again take the middle third. We conchue this process indefinitely. How many points of the segment OA will remain , undeleted ? At first'glance we might say that only the endpoints 0 and A will remain. This conclusion is supported, it would seem. by the following reasoning. We compute the sum of the lengths of all the segments deleted by the above process. (We recall that we took the".length of the en~re, segment O~ to be equal to 1.) At the first step we rejected i\ Segment of length 1/3, at the second step two segments of length 1/9, } at 'the third four segments of length 1/'J7, B.lld SQ ,op. The sum' of the lengtbs of all the deleted segments is equal to .':' ~

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Thus, the sum" of 'the lengths of the del~ted segments is exactly equal to the leagth of the original segment OA! And yet the above process leaves-bps ides 0 and A--an infinite number of undeleted points:To see this, we represeht each point of the unit segment OA by an infinite ternary-expansion. Each such representation consists of zeros, ones, and twos. We claim that the process of, deleting the" middle third" leaves behind exllctly those points which correspond to ternary expansions containing no ones (composed entirely 'of zeros and twos). I n the first step we deIeted the middle third of the unit interval, that is, those points which correspond to terna'ry expansions having a one in the first place. In the second step we deleted t~c middle third again. removing the expansions which have a'one in the , second place, and so forth. (Here we delete th~se points that can be represented by two ternary expansions if one of these expansions contains a one. For example, the endpoint of the first third of the line segment OA, the number, 1/3, can be represented by the ternary expan" sions . 0.1000 ...

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0.0222. '.:); this point we defete.) And so, the process leaves exactly those points which correspond to ternary exPansions consisting bnly of zeros and . twos. But there are infinitely many such numbers! Consequently. besides the endpoints, theN wilf still remain ipfiniteiy niany undeleted I points. F~1xam.PJe, ~he ;mint that corresponds to the representation )

0.020202 ...

(the ternary expansion of the number 1/4) will remain. The infinite ternary representation 0.020202... actually signifies the sum of the geometric progression

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By using the following geometric argument, we can persuade our~ selves that the point t 14 will not be deleted. The point 1/4 divides the whole interval [0, I] in a ratio of I : 3. After the removal of the segment [1/3. 2/3}, the point 1/4 remains in the half~open interval [0,1/3), which it divides in a ratio of 3: 1. After the second deletion it remains in the .open interval (2/9. 113), which it divides in a ratio of 3:·1, and so on. At no step will the point 1/4 be removed. I Tlius. it turns out that the process of deleting the" middle third" leads to a sct of points which, although it .. takes up no space at all" on the line segment (since the sum of the lengths of the deleted segments is equal, as we have made clear, to one), contains infinitely many points. This set of points possesses other interesting properties; however, to study them would. requite an exposition of concepts beyond the scope of our little book. Thus, we end here .

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· apeakInQ tMcheN and ItUdtnta . 101M of the baM tn&tbImaticaI literature of the Scviet Unlon. The teeN... are·lntanded ~ Introduce v_rf0U8.lIpectaOf mathetnatlcal thought and to enaaae thlltudent In . matf)ematlcaJ IclMty wbfch wnl fcet.r~ woric. Some of the lecturee provide an .a.mentaJy Introduction to. cerlaln· noneIementaIy topi", and othe.. ptMent mathetnatlcal . QOf1cept1 and IdeIIln g ...ter depth. The bQo1cs contain manY Ingenious problems with hints, IOlutlonl, and anawS$. A Ilgnlflcant feature fa the authors' · UI8 of new approach.. to make " complex ma1hematJcaI toplos accessible to both high school and college etudents. '"

inherent advantagel oVer Other

poulble Iyttema: Ita popul.rity . i. due to tHltoricaJ ar1d . biological, not mathematical . facto,.. In this~k, S. V. Fomfn dlacuues the origin. properties,

and applications of various number systems. Including the

. decimal. the bInary, and the ternary. His preuntlllfon offers

the student an introduction to mathematical abstraction and

then, through its examples, shows him the abstraction

at work.

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The University of Chicago Press

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JocJ W. Ttl/et and .. . . Thoma P. BlaMOiI The'mOlt common language of . numbers, the decimal system, haa not aJwaya been used . . unl~rsalry. From a purely ma$hentatical point of view,· . .the decimal ayatem "'" no

Paper

47

ISBN:

~22S-25669-3

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