Seeing there is annihilation (right able-bodied beloued Students in the Mathematickes) that is so alarming to Mathematicall practise, nor that doth added accost and arrest Calculators, afresh the Multiplications, Diuisions, aboveboard and cubical Extractions of abundant numbers, which besides the annoying expence of time, are for the best allotment subiect to abounding glace errors. I began appropriately to accede in my minde, by what certaine and accessible Art I ability remoue those hindrances.

— JOHN NEPAIR [NAPIER] (1616)

I do abhorrence sums. There is no greater aberration than to alarm accession an exact science. There are…hidden laws of Cardinal which it requires a apperception like abundance to perceive. For instance, if you add a sum from the basal up, and afresh afresh from the top down, the aftereffect is consistently different.

— M. P. LA TOUCHE (1878)

I cannot accept that anybody will crave multiplications at the amount of 40,000, or alike 4,000 per hour; such a apostle change as the octonary calibration should not be imposed aloft flesh in accepted for the annual of a few individuals.

— F. H. WALES (1936)

Most afterwards analysts accept no absorption in arithmetic.

— B. PARLETT (1979)

The arch purpose of this affiliate is to accomplish a accurate abstraction of the four basal processes of arithmetic: addition, subtraction, multiplication, and division. Abounding bodies attention accession as a atomic affair that accouchement apprentice and computers do, but we will see that accession is a alluring affair with abounding absorbing facets. It is important to accomplish a absolute abstraction of able methods for artful with numbers, back accession underlies so abounding computer applications.

Arithmetic is, in fact, a active accountable that has played an important allotment in the history of the world, and it still is ability accelerated development. In this chapter, we shall assay algorithms for accomplishing accession operations on abounding types of quantities, such as “floating point” numbers, acutely ample numbers, fractions (rational numbers), polynomials, and ability series; and we will additionally altercate accompanying capacity such as abject conversion, factoring of numbers, and the appraisal of polynomials.

The way we do accession is carefully accompanying to the way we represent the numbers we accord with, so it is acclimatized to activate our abstraction of the accountable with a altercation of the arch agency for apery numbers.

Positional characters application abject b (or abject b) is authentic by the rule

for example, . Our accepted decimal cardinal arrangement is, of course, the appropriate case back b is ten, and back the a’s are alleged from the “decimal digits” 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; in this case the subscript b in (1) may be omitted.

The simplest generalizations of the decimal cardinal arrangement are acquired back we booty b to be an accumulation greater than 1 and back we crave the a’s to be integers in the ambit 0 ≤ ak < b. This gives us the accepted bifold (b = 2), ternary (b = 3), quaternary (b = 4), quinary (b = 5), . . . cardinal systems. In general, we could booty b to be any nonzero number, and we could accept the a’s from any authentic set of numbers; this leads to some absorbing situations, as we shall see.

The dot that appears amid a0 and a–1 in (1) is alleged the abject point. (When b = 10, it is additionally alleged the decimal point, and back b = 2, it is sometimes alleged the bifold point, etc.) Continental Europeans about use a breach instead of a dot to denote the abject point; the English aforetime acclimated a aloft dot.

The a’s in (1) are alleged the digits of the representation. A chiffre ak for ample k is about said to be “more significant” than the digits ak for baby k; accordingly, the leftmost or “leading” chiffre is referred to as the best cogent chiffre and the rightmost or “trailing” chiffre is referred to as the atomic cogent digit. In the accepted bifold arrangement the bifold digits are about alleged bits; in the accepted hexadecimal arrangement (radix sixteen) the hexadecimal digits aught through fifteen are usually denoted by

The actual development of cardinal representations is a alluring story, back it parallels the development of acculturation itself. We would be action far afield if we were to appraise this history in minute detail, but it will be adorning to attending at its capital actualization here.

The age-old forms of cardinal representations, still begin in archaic cultures, are about based on groups of fingers, bags of stones, etc., usually with appropriate conventions about replacing a beyond accumulation or accumulation of, say, bristles or ten altar by one commodity of a appropriate affectionate or in a appropriate place. Such systems advance artlessly to the age-old agency of apery numbers in accounting form, as in the systems of Babylonian, Egyptian, Greek, Chinese, and Roman numerals; but such notations are analogously annoying for d accession operations except in the simplest cases.

During the twentieth century, historians of mathematics accept fabricated all-encompassing studies of aboriginal cuneiform tablets begin by archæologists in the Middle East. These studies actualization that the Babylonian bodies absolutely had two audible systems of cardinal representation: The numbers acclimated in acclimatized business affairs were accounting in a characters based on alignment by tens, hundreds, etc.; this characters was affiliated from advanced Mesopotamian civilizations, and ample numbers were hardly required. Back added difficult algebraic problems were considered, however, Babylonian mathematicians fabricated all-encompassing use of a agesimal (radix sixty) positional characters that was awful developed at atomic as aboriginal as 1750 B.C. This characters was altered in that it was absolutely a amphibian point anatomy of representation with exponents omitted; the able calibration agency or ability of sixty was to be supplied by the context, so that, for example, the numbers 2, 120, 7200, and were all accounting in an identical manner. The characters was abnormally acceptable for multiplication and division, application abetting tables, back radix-point alignment had no aftereffect on the answer. As examples of this Babylonian notation, accede the afterward excerpts from aboriginal tables: The aboveboard of 30 is 15 (which may additionally be read, “The aboveboard of is ”); the alternate of 81 = (1 21)60 is (44 26 40)60; and the aboveboard of the closing is (32 55 18 31 6 40)60. The Babylonians had a attribute for zero, but because of their “floating point” philosophy, it was acclimated alone aural numbers, not at the acclimatized end to denote a calibration factor. For the absorbing adventure of aboriginal Babylonian mathematics, see O. Neugebauer, The Exact Sciences in Antiquity (Princeton, N. J.: Princeton University Press, 1952), and B. L. van der Waerden, Science Awakening, translated by A. Dresden (Groningen: P. Noordhoff, 1954); see additionally D. E. Knuth, CACM 15 (1972), 671–677; 19 (1976), 108.

Fixed point positional characters was allegedly aboriginal conceived by the Maya Indians in axial America some 2000 years ago; their radix-20 arrangement was awful developed, abnormally in affiliation with ample annal and agenda dates. They began to use a accounting assurance for aught about A.D. 200. But the Spanish conquerors destroyed about all of the Maya books on history and science, so we accept analogously little ability about the amount of composure that built-in Americans had accomplished in arithmetic. Special-purpose multiplication tables accept been found, but no examples of assay are known. [See J. Eric S. Thompson, Contrib. to Amer. Anthropology and History 7 (Carnegie Inst. of Washington, 1941), 37–67; J. Justeson, “Pratiche di calcolo nell’antica mesoamerica,” Storia della Scienza 2 (Rome: Istituto della Enciclopedia Italiana, 2001), 976–990.]

Several centuries afore Christ, the Greek bodies active an aboriginal anatomy of the abacus to do their arithmetical calculations, application beach and/or dust on a lath that had rows or columns agnate in a acclimatized way to our decimal system. It is conceivably hasty to us that the aforementioned positional characters was never acclimatized to accounting forms of numbers, back we are so acclimatized to decimal reckoning with pencil and paper; but the greater affluence of artful by abacus (since autography was not a accepted skill, and back abacus users charge not acquire accession and multiplication tables) allegedly fabricated the Greeks feel it would be asinine alike to advance that accretion could be done bigger on “scratch paper.” At the aforementioned time Greek astronomers did accomplish use of a agesimal positional characters for fractions, which they had abstruse from the Babylonians.

Our decimal notation, which differs from the added age-old forms primarily because of its anchored abject point, calm with its attribute for aught to mark an abandoned position, was developed aboriginal in India aural the Hindu culture. The exact date back this characters aboriginal appeared is absolutely uncertain; about A.D. 600 seems to be a acceptable guess. Hindu science was awful developed at that time, decidedly in astronomy. The age-old accepted Hindu manuscripts that actualization decimal characters accept numbers accounting backwards (with the best cogent chiffre at the right), but anon it became accepted to put the best cogent chiffre at the left.

The Hindu attempt of decimal accession were brought to Persia about A.D. 750, as several important works were translated into Arabic; a arresting annual of this development is accustomed in a Hebrew certificate by Abraham Ibn Ezra, which has been translated into English in AMM 25 (1918), 99–108. Not affiliated afterwards this, al-Khwārizmī wrote his Arabic arbiter on the subject. (As acclaimed in Affiliate 1, our chat “algorithm” comes from al-Khwārizmī’s name.) His assignment was translated into Latin and was a able access on Leonardo Pisano (Fibonacci), whose book on accession (A.D. 1202) played a aloft role in the overextension of Hindu-Arabic numerals into Europe. It is absorbing to agenda that the left-to-right adjustment of autograph numbers was banausic during these two transitions, although Arabic is accounting from acclimatized to larboard while Hindu and Latin advisers about wrote from larboard to right. A abundant annual of the consecutive advancement of decimal numeration and accession into all genitalia of Europe during the aeon 1200–1600 has been accustomed by David Eugene Smith in his History of Mathematics 1 (Boston: Ginn and Co., 1923), Chapters 6 and 8.

Decimal characters was activated at aboriginal alone to accumulation numbers, not to fractions. Arabic astronomers, who appropriate fractions in their brilliant archive and added tables, affiliated to use the characters of Ptolemy (the acclaimed Greek astronomer), a characters based on agesimal fractions. This arrangement still survives today in our algebraic units of degrees, minutes, and seconds, and additionally in our units of time, as a balance of the aboriginal Babylonian agesimal notation. Aboriginal European mathematicians additionally acclimated agesimal fractions back ambidextrous with noninteger numbers; for example, Fibonacci gave the value

as an approximation to the abject of the blueprint x3 2×2 10x = 20. (The actual acknowledgment is 1◦ 22′ 7″ 42″′ 33IV 4V 38VI 30VII 50VIII 15IX 43X. . . .)

The use of decimal characters additionally for tenths, hundredths, etc., in a agnate way seems to be a analogously accessory change; but, of course, it is adamantine to breach with tradition, and agesimal fractions accept an advantage over decimal fractions because numbers such as can be bidding exactly, in a simple way.

Chinese mathematicians—who never acclimated agesimals—were allegedly the aboriginal bodies to assignment with the agnate of decimal fractions, although their actualization arrangement (lacking zero) was not originally a positional cardinal arrangement in the austere sense. Chinese units of weights and measures were decimal, so that Tsu Ch’ung-Chih (who died in A.D. 500 or 501) was able to accurate an approximation to π in the afterward form:

Here chang, . . . , hu are units of length; 1 hu (the bore of a cottony thread) equals 1/10 miao, etc. The use of such decimal-like fractions was adequately boundless in China afterwards about 1250.

An alpha anatomy of absolutely positional decimal fractions appeared in a 10th-century accession text, accounting in Damascus by an abstruse mathematician alleged al-Uqlīdisī (“the Euclidean”). He occasionally credible the abode of a decimal point, for archetype in affiliation with a botheration about admixture interest, the ciphering of 135 times (1.1)n for 1 ≤ n ≤ 5. [See A. S. Saidan, The Accession of al-Uqlīdisī (Dordrecht: D. Reidel, 1975), 110, 114, 343, 355, 481–485.] But he did not advance the abstraction actual fully, and his ambush was anon forgotten. Al-Samaw’al of Baghdad and Baku, autograph in 1172, accepted that but he had no acceptable way to address such approximations down. Several centuries anesthetized afore decimal fractions were reinvented by a Persian mathematician, al-Kāshī, who died in 1429. Al-Kāshī was a awful accomplished calculator, who gave the amount of 2π as follows, actual to 16 decimal places:

This was by far the best approximation to π accepted until Ludolph van Ceulen agilely affected 35 decimal places during the aeon 1586–1610.

Decimal fractions began to arise sporadically in Europe; for example, a alleged “Turkish method” was acclimated to compute 153.5 × 16.25 = 2494.375. Giovanni Bianchini developed them further, with applications to surveying, above-mentioned to 1450; but like al-Uqlīdisī, his assignment seems to accept had little influence. Christof Rudolff and François Viète appropriate the abstraction afresh in 1525 and 1579. Finally, an accession argument by Simon Stevin, who apart hit on the abstraction of decimal fractions in 1585, became popular. Stevin’s work, and the assay of logarithms anon afterwards, fabricated decimal fractions commonplace in Europe during the 17th century. [For added animadversion and references, see D. E. Smith, History of Mathematics 2 (1925), 228–247; V. J. Katz, A History of Mathematics (1993), 225–228, 345–348; and G. Rosińska, Quart. J. Hist. Sci. Tech. 40 (1995), 17–32.]

The bifold arrangement of characters has its own absorbing history. Abounding archaic tribes in actuality today are accepted to use a bifold or “pair” arrangement of counting (making groups of two instead of bristles or ten), but they do not adding in a accurate radix-2 system, back they do not amusement admiral of 2 in a appropriate manner. See The Diffusion of Counting Practices by Abraham Seidenberg, Univ. of Calif. Publ. in Math. 3 (1960), 215–300, for absorbing capacity about archaic cardinal systems. Accession “primitive” archetype of an about bifold arrangement is the accepted agreeable characters for cogent rhythms and durations of time.

Nondecimal cardinal systems were discussed in Europe during the seventeenth century. For abounding years astronomers had occasionally acclimated agesimal accession both for the accumulation and the apportioned genitalia of numbers, primarily back d multiplication [see John Wallis, Treatise of Algebra (Oxford: 1685), 18–22, 30]. The actuality that any accumulation greater than 1 could serve as abject was allegedly aboriginal declared in book by Blaise Pascal in De Numeris Multiplicibus, which was accounting about 1658 [see Pascal’s Œuvres Complètes (Paris: Éditions du Seuil, 1963), 84–89]. Pascal wrote, “Denaria enim ex instituto hominum, non ex necessitate naturæ ut vulgus arbitratur, et sane satis inepte, posita est”; i.e., “The decimal arrangement has been established, somewhat absurdly to be sure, according to man’s custom, not from a acclimatized call as best bodies think.” He declared that the duodecimal (radix twelve) arrangement would be a acceptable change, and he gave a aphorism for testing a duodecimal cardinal for divisibility by nine. Erhard Weigel approved to boom up action for the quaternary (radix four) arrangement in a alternation of publications alpha in 1673. A abundant altercation of radix-twelve accession was accustomed by Joshua Jordaine, Duodecimal Arithmetick (London: 1687).

Although decimal characters was about alone acclimated for accession during that era, added systems of weights and measures were rarely if anytime based on multiples of 10, and business affairs appropriate a acceptable accord of accomplishment in abacus quantities such as pounds, shillings, and pence. For centuries merchants had appropriately abstruse to compute sums and differences of quantities bidding in appropriate units of currency, weights, and measures; appropriately they were accomplishing accession in nondecimal cardinal systems. The accepted units of aqueous admeasurement in England, dating from the 13th aeon or earlier, are decidedly noteworthy:

Quantities of aqueous bidding in gallons, pottles, quarts, pints, etc. were about accounting in bifold notation. Conceivably the accurate inventors of bifold accession were British wine merchants!

The aboriginal accepted actualization of authentic bifold characters was about 1605 in some abstruse manuscripts of Thomas Harriot (1560–1621). Harriot was a artistic man who aboriginal became acclaimed by advancing to America as a adumbrative of Sir Walter Raleigh. He invented (among added things) a characters like that now acclimated for “less than” and “greater than” relations; but for some acumen he chose not to broadcast abounding of his discoveries. Excerpts from his addendum on bifold accession accept been reproduced by John W. Shirley, Amer. J. Physics 19 (1951), 452–454; Harriot’s assay of bifold characters was aboriginal cited by Frank Morley in The Accurate Monthly 14 (1922), 60–66.

The aboriginal arise assay of the bifold arrangement appeared in the assignment of a arresting Cistercian bishop, Juan Caramuel de Lobkowitz, Mathesis Biceps 1 (Campaniæ: 1670), 45–48. Caramuel discussed the representation of numbers in radices 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, and 60 at some length, but gave no examples of accession operations in nondecimal systems except in the agesimal case.

Ultimately, an commodity by G. W. Leibniz [Mémoires de l’Académie Royale des Sciences (Paris, 1703), 110–116], which illustrated bifold addition, subtraction, multiplication, and division, absolutely brought bifold characters into the limelight, and his commodity is usually referred to as the bearing of radix-2 arithmetic. Leibniz afterwards referred to the bifold arrangement absolutely frequently. He did not acclaim it for activated calculations, but he fatigued its accent in number-theoretical investigations, back patterns in cardinal sequences are about added credible in bifold characters than they are in decimal; he additionally saw a mystical acceptation in the actuality that aggregate is expressible in agreement of aught and one. Leibniz’s abstruse manuscripts actualization that he had been absorbed in bifold characters as aboriginal as 1679, back he referred to it as a “bimal” arrangement (analogous to “decimal”).

A accurate abstraction of Leibniz’s aboriginal assignment with bifold numbers has been fabricated by Hans J. Zacher, Die Hauptschriften zur Dyadik von G. W. Leibniz (Frankfurt am Main: Klostermann, 1973). Zacher credibility out that Leibniz was accustomed with John Napier’s alleged “local arithmetic,” a way for artful with stones that amounts to application a radix-2 abacus. [Napier had arise the abstraction of bounded accession as allotment three of his little book Rabdologiæ in 1617; it may be alleged the world’s aboriginal “binary computer,” and it is absolutely the world’s cheapest, although Napier acquainted that it was added agreeable than practical. See Martin Gardner’s altercation in Knotted Doughnuts and Added Algebraic Entertainments (New York: Freeman, 1986), Affiliate 8.]

It is absorbing to agenda that the important abstraction of abrogating admiral to the acclimatized of the abject point was not yet able-bodied accepted at that time. Leibniz asked James Bernoulli to annual π in the bifold system, and Bernoulli “solved” the botheration by demography a 35-digit approximation to π, adding it by 1035, and afresh cogent this accumulation in the bifold arrangement as his answer. On a abate calibration this would be like adage that π ≈ 3.14, and (314)10 = (100111010)2; appropriately π in bifold is 100111010! [See Leibniz, Math. Schriften, edited by C. I. Gerhardt, 3 (Halle: 1855), 97; two of the 118 $.25 in the acknowledgment are incorrect, due to computational errors.] The motive for Bernoulli’s adding was allegedly to see whether any simple arrangement could be empiric in this representation of π.

Charles XII of Sweden, whose aptitude for mathematics conceivably exceeded that of all added kings in the history of the world, hit on the abstraction of radix-8 accession about 1717. This was allegedly his own invention, although he had met Leibniz briefly in 1707. Charles acquainted that abject 8 or 64 would be added acceptable for adding than the decimal system, and he advised introducing octal accession into Sweden; but he died in action afore aldermanic such a change. [See The Works of Voltaire 21 (Paris: E. R. DuMont, 1901), 49; E. Swedenborg, Gentleman’s Magazine 24 (1754), 423–424.]

Octal characters was proposed additionally in colonial America afore 1750, by the Rev. Hugh Jones, assistant at the College of William and Mary [see Gentleman’s Magazine 15 (1745), 377–379; H. R. Phalen, AMM 56 (1949), 461–465].

More than a aeon later, a arresting Swedish-American civilian architect alleged John W. Nystrom absitively to backpack Charles XII’s affairs a footfall further, by devising a complete arrangement of numeration, weights, and measures based on radix-16 arithmetic. He wrote, “I am not afraid, or do not hesitate, to apostle a bifold arrangement of accession and metrology. I apperceive I accept attributes on my side; if I do not accomplish to affect aloft you its annual and abundant accent to mankind, it will reflect that abundant beneath acclaim aloft our generation, aloft our accurate men and philosophers.” Nystrom devised appropriate agency for pronouncing hexadecimal numbers; for example, (C0160)16 was to be apprehend “vybong, bysanton.” His absolute arrangement was alleged the Tonal System, and it is declared in J. Franklin Inst. 46 (1863), 263–275, 337–348, 402–407. A agnate system, but application abject 8, was formed out by Alfred B. Taylor [Proc. Amer. Pharmaceutical Assoc. 8 (1859), 115–216; Proc. Amer. Philosophical Soc. 24 (1887), 296–366]. Increased use of the French (metric) arrangement of weights and measures prompted all-encompassing agitation about the affirmation of decimal accession during that era; indeed, octal accession was alike actuality proposed in France [J. D. Collenne, Le Système Octaval (Paris: 1845); Aimé Mariage, Numération par Huit (Paris: Le Nonnant, 1857)].

The bifold arrangement was able-bodied accepted as a concern anytime back Leibniz’s time, and about 20 aboriginal references to it accept been aggregate by R. C. Archibald [AMM 25 (1918), 139–142]. It was activated chiefly to the adding of powers, as explained in Area 4.6.3, and to the assay of assertive amateur and puzzles. Giuseppe Peano [Atti della R. Accademia delle Scienze di Torino 34 (1898), 47–55] acclimated bifold characters as the abject of a “logical” actualization set of 256 syms. Joseph Bowden [Special Capacity in Theoretical Accession (Garden City: 1936), 49] gave his own arrangement of classification for hexadecimal numbers.

The book History of Bifold and Added Nondecimal Numeration by Anton Glaser (Los Angeles: Tomash, 1981) contains an advisory and about complete altercation of the development of bifold notation, including English translations of abounding of the works cited aloft [see Historia Math. 10 (1983), 236–243].

Much of the contempo history of cardinal systems is affiliated with the development of artful machines. Charles Babbage’s notebooks for 1838 actualization that he advised application nondecimal numbers in his Analytical Engine [see M. V. Wilkes, Historia Math. 4 (1977), 421]. Increased absorption in automated accessories for arithmetic, abnormally for multiplication, led several bodies in the 1930s to accede the bifold arrangement for this purpose. A decidedly adorable annual of such action appears in the commodity “Binary Calculation” by E. William Phillips [Journal of the Institute of Actuaries 67 (1936), 187–221] calm with a almanac of the altercation that followed a address he gave on the subject. Phillips began by saying, “The ultimate aim [of this paper] is to actuate the accomplished affable apple to carelessness decimal numeration and to use octonal [that is, abject 8] numeration in its place.”

Modern readers of Phillips’s commodity will conceivably be afraid to ascertain that a radix-8 cardinal arrangement was appropriately referred to as “octonary” or “octonal,” according to all dictionaries of the English accent at that time, aloof as the radix-10 cardinal arrangement is appropriately alleged either “denary” or “decimal”; the chat “octal” did not arise in English accent dictionaries until 1961, and it allegedly originated as a appellation for the abject of a assertive chic of exhaustion tubes. The chat “hexadecimal,” which has crept into our accent alike added recently, is a admixture of Greek and Latin stems; added able agreement would be “senidenary” or “sedecimal” or alike “adecimal,” but the closing is conceivably too risqué for computer programmers.

The animadversion by Mr. Wales that is quoted at the alpha of this affiliate has been taken from the altercation printed with Phillips’s paper. Accession man who abounding the aforementioned address objected to the octal arrangement for business purposes: “5% becomes per 64, which sounds rather horrible.”

Phillips got the afflatus for his proposals from an cyberbanking ambit that was able of counting in bifold [C. E. Wynn-Williams, Proc. Roy. Soc. London A136 (1932), 312–324]. Electromechanical and cyberbanking chip for accepted accession operations was developed during the backward 1930s, conspicuously by John V. Atanasoff and George R. Stibitz in the U.S.A., L. Couffignal and R. Valtat in France, Helmut Schreyer and Konrad Zuse in Germany. All of these inventors acclimated the bifold system, although Stibitz afterwards developed excess-3 binary-coded-decimal notation. A alluring annual of these aboriginal developments, including reprints and translations of important a documents, appears in Brian Randell’s book The Origins of Digital Computers (Berlin: Springer, 1973).

The aboriginal American accelerated computers, congenital in the aboriginal 1940s, acclimated decimal arithmetic. But in 1946, an important announcement by A. W. Burks, H. H. Goldstine, and J. von Neumann, in affiliation with the architecture of the aboriginal stored-program computers, gave abundant affidavit for authoritative a aitionist abandonment from attitude and application base-two characters [see John von Neumann, Collected Works 5, 41–65]. Back afresh bifold computers accept multiplied. Afterwards a dozen years of acquaintance with bifold machines, a altercation of the about advantages and disadvantages of radix-2 characters was accustomed by W. Buchholz in his cardboard “Fingers or Fists?” [CACM 2, 12 (December 1959), 3–11].

The MIX computer acclimated in this book has been authentic so that it can be either bifold or decimal. It is absorbing to agenda that about all MIX programs can be bidding afterwards alive whether bifold or decimal characters is actuality used—even back we are accomplishing calculations involving multiple-precision arithmetic. Appropriately we acquisition that the best of abject does not decidedly access computer programming. (Noteworthy exceptions to this statement, however, are the “Boolean” algorithms discussed in Area 7.1; see additionally Algorithm 4.5.2B.)

There are several altered agency to represent abrogating numbers in a computer, and this sometimes influences the way accession is done. In adjustment to accept these notations, let us aboriginal accede MIX as if it were a decimal computer; afresh anniversary chat contains 10 digits and a sign, for example

This is alleged the active consequence representation. Such a representation agrees with accepted notational conventions, so it is adopted by abounding programmers. A abeyant disadvantage is that bare aught and additional aught can both be represented, while they usually should beggarly the aforementioned number; this achievability requires some affliction in practice, although it turns out to be advantageous at times.

Most automated calculators that do decimal accession use accession arrangement alleged ten’s accompaniment notation. If we decrease 1 from 00000 00000, we get 99999 99999 in this notation; in added words, no absolute assurance is absorbed to the number, and adding is done modulo 1010. The cardinal −12345 67890 would arise as

in ten’s accompaniment notation. It is accepted to attention any cardinal whose arch chiffre is 5, 6, 7, 8, or 9 as a abrogating amount in this notation, although with account to accession and accession there is no abuse in apropos (3) as the cardinal 87654 32110 if it is acceptable to do so. Apprehension that there is no botheration of bare aught in such a system.

The aloft aberration amid active consequence and ten’s accompaniment notations in convenance is that alive acclimatized does not bisect the consequence by ten; for example, the cardinal –11 = . . . 99989, confused acclimatized one, gives . . . 99998 = –2 (assuming that a about-face to the acclimatized inserts “9” as the arch chiffre back the cardinal confused is negative). In general, x confused acclimatized one chiffre in ten’s accompaniment characters will accord ⌊x/10⌋, whether x is absolute or negative.

A accessible disadvantage of the ten’s accompaniment arrangement is the actuality that it is not symmetric about zero; the p-digit abrogating cardinal 500 . . . 0 is not the abrogating of any p-digit absolute number. Appropriately it is accessible that alteration x to –x will account overflow. (See challenge 7 and 31 for a altercation of radix-complement characters with absolute precision.)

Another characters that has been acclimated back the age-old canicule of accelerated computers is alleged nines’ accompaniment representation. In this case the cardinal −12345 67890 would arise as

Each chiffre of a abrogating cardinal (–x) is according to 9 bare the agnate chiffre of x. It is not difficult to see that the nines’ accompaniment characters for a abrogating cardinal is consistently one beneath than the agnate ten’s accompaniment notation. Accession and accession are done modulo 1010 – 1, which agency that a backpack off the larboard end is to be added at the acclimatized end. (See the altercation of accession modulo w − 1 in Area 3.2.1.1.) Afresh there is a abeyant botheration with bare zero, back 99999 99999 and 00000 00000 denote the aforementioned value.

The account aloof explained for radix-10 accession administer in a agnate way to radix-2 arithmetic, area we accept active magnitude, two’s complement, and ones’ accompaniment notations. Two’s accompaniment accession on n-bit numbers is accession modulo 2n; ones’ accompaniment accession is modulo 2n − 1. The MIX computer, as acclimated in the examples of this chapter, deals alone with active consequence arithmetic; however, accession procedures for accompaniment notations are discussed in the accompanying argument back it is important to do so.

Detail-oriented readers and archetype editors should apprehension the position of the apostrophe in agreement like “two’s complement” and “ones’ complement”: A two’s accompaniment cardinal is complemented with account to a distinct ability of 2, while a ones’ accompaniment cardinal is complemented with account to a affiliated arrangement of 1s. Indeed, there is additionally a “twos’ accompaniment notation,” which has abject 3 and complementation with account to (2 . . . 22)3.

Descriptions of apparatus accent about acquaint us that a computer’s chip is set up with the abject point at a accurate abode aural anniversary numeric word. Such statements should usually be disregarded. It is bigger to apprentice the rules apropos area the abject point will arise in the aftereffect of an apprenticeship if we accept that it lies in a assertive abode beforehand. For example, in the case of MIX we could attention our operands either as integers with the abject point at the acute right, or as fractions with the abject point at the acute left, or as some admixture of these two extremes; the rules for the actualization of the abject point afterwards addition, subtraction, multiplication, or assay are straightforward.

It is accessible to see that there is a simple affiliation amid abject b and abject bk:

where

see exercise 8. Appropriately we accept simple techniques for converting at afterimage between, say, bifold and hexadecimal notation.

Many absorbing variations on positional cardinal systems are accessible in accession to the accepted b-ary systems discussed so far. For example, we ability accept numbers in abject (–10), so that

Here the alone digits amuse 0 ≤ ak ≤ 9 aloof as in the decimal system. The cardinal 12345 67890 appears in the “negadecimal” arrangement as

since the closing represents 10305070900 – 9070503010. It is absorbing to agenda that the abrogating of this number, –12345 67890, would be written

and, in fact, every absolute cardinal whether absolute or abrogating can be represented afterwards a assurance in the –10 system.

Negative-base systems were aboriginal advised by Vittorio Grünwald [Giornale di Matematiche di Battaglini 23 (1885), 203–221, 367], who explained how to accomplish the four accession operations in such systems; Grünwald additionally discussed abject extraction, divisibility tests, and abject conversion. However, his assignment seems to accept had no aftereffect on added research, back it was arise in a rather abstruse journal, and it was anon forgotten. The aing advertisement about negative-base systems was allegedly by A. J. Kempner [AMM 43 (1936), 610–617], who discussed the backdrop of noninteger radices and remarked in a comment that abrogating radices would be achievable too. Afterwards twenty added years the abstraction was rediscovered again, this time by Z. Pawlak and A. Wakulicz [Bulletin de l’Académie Polonaise des Sciences, Classe III, 5 (1957), 233–236; Série des sciences techniques 7 (1959), 713–721], and additionally by L. Wadel [IRE Affairs EC-6 (1957), 123]. Beginning computers alleged SKRZAT 1 and BINEG, which acclimated –2 as the abject of arithmetic, were congenital in Poland in the backward 1950s; see N. M. Blachman, CACM 4 (1961), 257; R. W. Marczyński, Ann. Hist. Accretion 2 (1980), 37–48. For added references see IEEE Affairs EC-12 (1963), 274–277; Computer Architecture 6 (May 1967), 52–63. There is affirmation that the abstraction of abrogating bases occurred apart to absolutely a few people. For example, D. E. Knuth had discussed negative-radix systems in 1955, calm with a added generalization to complex-valued bases, in a abbreviate cardboard submitted to a “science aptitude search” challenge for high-school seniors.

The abject 2i gives acceleration to a arrangement alleged the “quater-imaginary” cardinal arrangement (by affinity with “quaternary”), which has the abnormal affection that every circuitous cardinal can be represented with the digits 0, 1, 2, and 3 afterwards a sign. [See D. E. Knuth, CACM 3 (1960), 245–247; 4 (1961), 355.] For example,

Here the cardinal (a2n . . . a1a0.a–1 . . . a–2k)2i is according to

so about-face to and from quater-imaginary characters reduces to about-face to and from abrogating quaternary representation of the absolute and abstract parts. The absorbing acreage of this arrangement is that it allows multiplication and assay of circuitous numbers to be done in a adequately unified address afterwards alleviative absolute and abstract genitalia separately. For example, we can accumulate two numbers in this arrangement abundant as we do with any base, alone application a altered backpack rule: Whenever a chiffre exceeds 3 we decrease 4 and backpack –1 two columns to the left; back a chiffre is negative, we add 4 to it and backpack 1 two columns to the left. The afterward archetype shows this appropriate backpack aphorism at work:

A agnate arrangement that uses aloof the digits 0 and 1 may be based on , but this requires an absolute nonrepeating amplification for the simple cardinal “i” itself. Vittorio Grünwald proposed application the digits 0 and in odd-numbered positions, to abstain such a problem; but that absolutely boodle the accomplished arrangement [see Commentari dell’Ateneo di Brescia (1886), 43–54].

Another “binary” circuitous cardinal arrangement may be acquired by application the abject i − 1, as appropriate by W. Penney [JACM 12 (1965), 247–248]:

In this system, alone the digits 0 and 1 are needed. One way to authenticate that every circuitous cardinal has such a representation is to accede the absorbing set S apparent in Fig. 1; this set is, by definition, all credibility that can be accounting as ∑k≥1ak(i − 1)−k, for an absolute arrangement a1, a2, a3, . . . of zeros and ones. It is additionally accepted as the “twindragon fractal” [see M. F. Barnsley, Fractals Everywhere, additional copy (Academic Press, 1993), 306, 310]. Figure 1 shows that S can be addle into 256 pieces coinciding to S. Apprehension that if the diagram of S is rotated counterclockwise by 135°, we access two adjoining sets coinciding to () S, because (i − 1)S = S ∪ (S 1). For capacity of a affidavit that S contains all circuitous numbers that are of abundantly baby magnitude, see exercise 18.

Fig. 1. The fractal set S alleged the “twindragon.”

Perhaps the prettiest cardinal arrangement of all is the counterbalanced ternary notation, which consists of radix-3 representation application –1, 0, and 1 as “trits” (ternary digits) instead of 0, 1, and 2. If we let the attribute angle for –1, we accept the afterward examples of counterbalanced ternary numbers:

One way to acquisition the representation of a cardinal in the counterbalanced ternary arrangement is to alpha by apery it in accustomed ternary notation; for example,

(A actual simple pencil-and-paper adjustment for converting to ternary characters is accustomed in exercise 4.4–12.) Now add the absolute cardinal . . . 11111.11111 . . . in ternary notation; we obtain, in the archetype above, the absolute number

Finally, decrease . . . 11111.11111 . . . by decrementing anniversary digit; we get

This action may acutely be fabricated accurate if we alter the bogus absolute cardinal . . . 11111.11111 . . . by a cardinal with appropriately abounding ones.

The counterbalanced ternary cardinal arrangement has abounding affable properties:

Addition in the counterbalanced ternary arrangement is absolutely simple, application the table

(The three inputs to the accession are the digits of the numbers to be added and the backpack digit.) Accession is antithesis followed by addition. Multiplication additionally reduces to antithesis and addition, as in the afterward example:

Representation of numbers in the counterbalanced ternary arrangement is around present in a acclaimed algebraic puzzle, frequently alleged “Bachet’s botheration of weights”—although it was already declared by Fibonacci four centuries afore Bachet wrote his book, and by abarī in Persia added than 100 years afore Fibonacci. [See W. Ahrens, Mathematische Unterhaltungen und Spiele 1 (Leipzig: Teubner, 1910), Area 3.4; H. Hermelink, Janus 65 (1978), 105–117.] Positional cardinal systems with abrogating digits were invented by J. Colson [Philos. Trans. 34 (1726), 161–173], afresh abandoned and rediscovered about 100 years afterwards by Sir John Leslie [The Aesthetics of Accession (Edinburgh: 1817); see pages 33–34, 54, 64–65, 117, 150], and by A. Cauchy [Comptes Rendus Acad. Sci. 11 (Paris, 1840), 789–798]. Cauchy acicular out that abrogating digits accomplish it accidental for a actuality to acquire the multiplication table accomplished 5 × 5. A affirmation that such cardinal systems were accepted in India affiliated ago [J. Bharati, Vedic Mathematics (Delhi: Motilal Banarsidass, 1965)] has been refuted by K. S. Shukla [Mathematical Education 5, 3 (1989), 129–133]. The aboriginal accurate actualization of “pure” counterbalanced ternary characters was in an commodity by Léon Lalanne [Comptes Rendus Acad. Sci. 11 (Paris, 1840), 903–905], who was a artist of automated accessories for arithmetic. Thomas Fowler apart invented and complete a counterbalanced ternary calculator at about the aforementioned time [see Report British Assoc. Adv. Sci. 10 (1840), 55; 11 (1841), 39–40]. The counterbalanced ternary cardinal arrangement was mentioned alone rarely for the aing 100 years, until the development of the aboriginal cyberbanking computers at the Moore School of Electrical Engineering in 1945–1946; at that time it was accustomed austere application as a accessible backup for the decimal system. The complication of accession chip for counterbalanced ternary accession is not abundant greater than it is for the bifold system, and a accustomed cardinal requires alone ln 2/ ln 3 ≈ 63% as abounding chiffre positions for its representation. Discussions of the counterbalanced ternary arrangement arise in AMM 57 (1950), 90–93, and in Accelerated Accretion Devices, Engineering Analysis Associates (McGraw–Hill, 1950), 287–289. The beginning Russian computer SETUN was based on counterbalanced ternary characters [see CACM 3 (1960), 149–150], and conceivably the symmetric backdrop and simple accession of this cardinal arrangement will prove to be absolutely important someday—when the “flip-flop” is replaced by a “flip-flap-flop.”

Positional characters generalizes in accession important way to a mixed-radix system. Accustomed a arrangement of numbers 〈bn〉 (where n may be negative), we define

In the simplest mixed-radix systems, we assignment alone with integers; we let b0, b1, b2, . . . be integers greater than one, and accord alone with numbers that accept no abject point, area an is appropriate to lie in the ambit 0 ≤ an < bn.

One of the best important mixed-radix systems is the factorial cardinal system, area bn = n 2. Application this system, which was accepted in 13th-century India, we can represent every absolute accumulation abnormally in the form

where 0 ≤ ck ≤ k for 1 ≤ k ≤ n, and cn ≠ 0. (See Algorithms 3.3.2P and 3.4.2P.)

Mixed-radix systems are accustomed in acclimatized life, back we accord with units of measure. For example, the abundance “3 weeks, 2 days, 9 hours, 22 minutes, 57 seconds, and 492 milliseconds” is according to

The abundance “10 pounds, 6 shillings, and thruppence ha’penny” was already according to pence in British currency, afore Abundant Britain afflicted to a absolutely decimal budgetary system.

It is accessible to add and decrease mixed-radix numbers by application a aboveboard generalization of the accepted accession and accession algorithms, provided of advance that the aforementioned mixed-radix arrangement is actuality acclimated for both operands (see exercise 4.3.1–9). Similarly, we can calmly accumulate or bisect a mixed-radix cardinal by baby accumulation constants, application simple extensions of the accustomed pencil-and-paper methods.

Mixed-radix systems were aboriginal discussed in abounding generality by Georg Cantor [Zeitschrift für Math. und Physik 14 (1869), 121–128]. Challenge 26 and 29 accord added advice about them.

Several questions apropos aberrant radices accept been advised by W. Parry, Acta Math. Acad. Sci. Hung. 11 (1960), 401–416.

Besides the systems declared in this section, several added agency to represent numbers are mentioned abroad in this alternation of books: the combinatorial cardinal arrangement (exercise 1.2.6–56); the Fibonacci cardinal arrangement (exercises 1.2.8–34, 5.4.2–10); the phi cardinal arrangement (exercise 1.2.8–35); modular representations (Section 4.3.2); Gray cipher (Section 7.2.1); and Roman numerals (Section 9.1).

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