NUMBER SYSTEM

any of various sets of symbols and the rules for using them to express quantities as the basis for counting, comparing amounts, performing calculations, determining order, making measurements, representing value, setting limits, abstracting quantities, coding information, and transmitting data. The most elementary representation of numbers is the tally or unitary system of notation. It originally consisted of single strokes that were put into one-to-one correspondence with the items being counted; later these were combined into groups of five or more. The origins of our modern decimal, or base-10, number system can be traced to ancient Egyptian, Babylonian (Sumerian), and Chinese roots. The bulk of the credit for the base-10 system goes to the Hindu-Arabic mathematicians of the 8th to 11th centuries AD, and the beginning of our modern notation is attributed to the work Liber abaci published by Leonardo of Pisa (Fibonacci) in AD 1202. The early Egyptians used a base-10 system that had a different symbol for each power of 10 up to 106, but it lacked a place-value notation and an explicit number zero. Similar systems were used by the ancient Chinese, Cretans, Greeks, Hebrews, and Romans. On the other hand, the Babylonian system was an incomplete sexagesimal (base-60) positional notation. It used only two symbols instead of the 60 distinct ones that a base-60 system could use, and thus suffered from ambiguities in representing value that could be resolved only by analyzing the context. The Mayan system was nominally vigesimal (based on 20), but to conform to their 360-day calendar it was modified so that successive positions had the values: 200 = 1, 201 = 20, 18 201 = 360, 18 202 = 7,200, 18 203 = 144,000, etc., rather than strict powers of 20 (200 = 1, 201 = 20, 202 = 400, 203 = 8,000, 204 = 160,000, . . . ). The use of zero as a numeral appeared sporadically in Egyptian number systems. It was used, however, only between two numbers to indicate an empty position, never at the end of a number. While the early Chinese did not have a symbol for zero, the invention and use of their abacus suggests that they had an implied appreciation for positional base notation and zero as a number. The Mayans did have a zero symbol, but their inconsistency in base notation rendered it virtually useless for computations. The first use of zero as a place holder in positional base notation was due probably to Muhammad ibn Musa al-Khwarizmi (c. 780850). This use of zero and the use of western Arabic (Gobar) numerals were spread throughout Europe in the 10th century principally by the efforts of Gerbert, who later became Pope Sylvester II. The requirements of more demanding measurements, analytical quantifications, and complex calculations provided the impetus for the transfer from mere symbolic representation of amounts to modern number systems. The distinguishing characteristics of a modern number system are its use of base position notation (place value), zero as a number, and a point or comma to separate the parts of numbers greater and less than 1. Some of the ancient number systems employed positional (base) notation but in an inconsistent and often confusing way. Since a separate symbol exists for each element within a modern number system, and its position determines its weighted value, greater condensation and simplicity is possible in representing numbers. This principle also makes it easy to extend calculations governed by general principles to other problems. The use of zero as a number permits the exact alignment of numbers for calculations and provides a consistent means of representing them. Before the adoption of positional base notation, zero, and the point, calculations such as multiplication, division, and root extraction had to be relegated to a handful of experts. By the 1100s the algorists, using base-10 notation, were successfully challenging the abacists (those using the abacus) in the speed and accuracy of calculations and had the advantage of a permanent written record of their results. The development and widespread use of a number system with these components greatly enhanced the precision and ease of calculations needed in fields such as astronomy, manufacturing, and navigation. It eventually led to even more efficient forms of handling data such as logarithms, slide rules, mechanical and electrical calculators, and computers. A general representation of numbers in any base may be expressed in the following way: cnbn + cn - 1bn - 1 + . . . c2b2 + c1b1 + c0b0 d1b-1 + d2b-2 + . . . + dnb-n, where b represents the base, and ci and di are place-value coefficients. The number represented by the preceding extended expression would ordinarily be written without the plus signs and the powers of the base (these are implied by the place-value property of the system: cncn - 1 . . . c2c1c0d1d2 . . . dn). In this representation: (1) each successive digit, counting leftward from the point, is the coefficient of an ascending power of the base beginning with b0 (that is, 1); (2) the successive digits, counting to the right of the point, are the coefficients of descending powers of the base, beginning with b-1; (3) numbers can be directly aligned around their basimal points, and calculations can be performed directly without having to determine how the order of the symbols affects their values (e.g., LXIV = 64, while XLVI = 46); (4) the number of symbols required equals the value of the base (e.g., in base 6, there are 6 symbols: 0, 1, 2, 3, 4, 5); (5) the symbols begin with zero and end with a value one less than the base (0, 1, 2, . . . b - 1); (6) the base does not have a separate symbol but is always represented by 10 (one times b1 plus zero times b0) so that 10 equals 2 in base 2, 12 in base 12, etc.; (7) the value of a number is equal to the sum of the products of the coefficients multiplied by their respective base powers. In the base-10 number system, the number 983.75 is a compact way of writing 900 + 80 + 3 + 7/10 + 5/100, or 9 102 + 8 101 + 3 100 + 7 10-1 + 5 10-2. For comparison, the number 983.75 is expressed in base 2 as 1111010111.11 (that is, 1 29 + 1 28 + 1 27 + 1 26 + 1 24 + 1 22 + 1 21 + 1 100 + 1 2-1 + 1 2-2) and in base 8 as 1727.6 (that is, 1 83 + 7 82 + 2 81 + 7 80 + 6 8-1). Note that only in base 10 do the coefficients translate directly to numerical value without requiring a multiplication. Two properties illustrated by the previous example are: (1) a rational number (expressible as a ratio of integers) remains a rational number when expressed in any integral base; (2) any rational number can be represented by a terminating or repeating basimal (decimal in base 10). Thomas Harriot (15601621) was the first to give a generalized treatment of positional number systems; but since his work was unpublished much of the credit is attributed to Gottfried Leibniz (16461716), who developed it independently. Leibniz was a great advocate of the base-2 systemfor him, 1 stood for God and 0 the void. The base-2 (binary, or dyadic) number system is ideal for reliable compact digital storage. Whereas the decimal system requires 10 symbols (0, 1, 2, . . . 9) and thus 10 electrical or mechanical states to represent each digit, the binary system has only two symbols, 0 and 1, and thus only requires two electrical states, most reliably off and on, to represent each digit. To the human eye, a number like 1,866 in base 10 appears more compact than the binary equivalent 11101001010. But since 10 states are required per digit in base 10, a 4-digit number requires 10 10 10 10 = 10,000 states, while the binary representation only requires 211 = 2,048 states. Numbers can be classified in many ways. The simplest class comprises the counting, or natural, numbers (1, 2, 3, . . . ), which with the addition of zero (not widely used until the 13th century) are known as the whole numbers. The use of the negatives of these numbers (-1, -2, -3) and negative numbers in general was not widely accepted until the 17th century. The natural numbers, their negative equivalents, and the zero constitute the integers. Rational numbers (fractions) date back to the earliest number systems; they can be defined as numbers expressible as ratios of integers, or equivalently, as terminating, or repeating, decimals (4 = 4/1; 3 1/7 = 22/7 = 3.142857 . . . ; 2 1/2 = 2.5; 1/3 = .333 . . . ). Irrational numbers cannot be expressed as a ratio of integers or as decimals that terminate or repeat; examples of them are = 1.4142 . . . ; p = 3.14159 . . . ; and e = 2.71828 . . . . Taken together, the rational and irrational numbers encompass the real number (q.v.) system. Numbers that are the even roots of negative numbers (, 4, 6, etc.) are not members of the real-number system; such numbers are called imaginary. They were first identified by Hero of Alexandria (c. AD 62), but up to the mid-19th century they were, for the most part, ignored or considered meaningless. The basic unit of the imaginary-number system is the irreducible form , which is represented by i. Numbers that have a real and an imaginary component are known as complex numbers. Complex numbers are represented by the form a + bi; a is the real part and bi the imaginary part. Thus, the reals (for which b = 0) or the imaginaries (a = 0), taken by themselves, are subsets of the complex-number system. The magnitudes and signs of purely real or purely imaginary numbers may be conveniently represented by one-dimensional graphs, that is, as points along a line or as distancespositive or negativebetween those points and a reference point corresponding to the number zero (the origin). Complex numbers in general, however, require two dimensions: each complex number corresponds to a point in a plane or to a line segment (a vector) directed from the origin to the point. These two-dimensional graphs of complex numbers were introduced independently about 1800 by Caspar Wessel of Norway and Jean Argand of France. The Irish mathematician Sir William Hamilton (180565), by dropping the commutative axiom of multiplication (ab = ba), developed a system of hypercomplex numbers that he called quaternions, which have a basis of four elements (1,i,j,k). The quaternions proved adaptable to operations involving vectors in three-dimensional space, but nowadays they are regarded mostly as historical antecedents to the form of vector analysis developed by the American scientist J. Willard Gibbs. The numberlike quantities represented by multidimensional expressions now form the currency of several branches of mathematical study. See also algebra; matrix; tensor analysis; vector analysis.