Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and simplest branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term higher arithmeticDavenport, Harold (1999). The Higher Arithmetic: An Introduction to the Theory of Numbers (7th ed.). Cambridge, England: Cambridge University Press. ISBN 0521634466. as a synonym for number theory, but this should not be confused with elementary arithmetic.

History

The prehistory of arithmetic is limited by a very small number of artifacts indicating a clear conception of addition and subtraction, the most well known being the Ishango Bone from Africa, dating from 18,000 BCE.

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic circa 1850 BCE, historians can only infer the methods utilized to generate the arithmetical results (see Plimpton 322). Likewise, a definitive algorithm for multiplication and the use of unit fractions can be found in the Rhind Mathematical Papyrus dating from Ancient Egypt circa 1650 BCE.

In the Pythagorean school, in the second half of the 6th century BCE, arithmetic was considered one of the four quantitative or mathematical sciences (Mathemata). These were carried over in mediæval universities as the Quadrivium which, together with the Trivium of grammar, rhetoric and dialectic, constituted the septem liberales artes (seven liberal arts).

Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Arabic numerals and decimal place notation for numbers. Although it is now considered elementary, its simplicity is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician Archimedes devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.

Decimal Arithmetic

Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,...,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: for example, 507.36 denotes 5 hundreds (10²), plus 0 tens (10¹), plus 7 units (10⁰), plus 3 tenths (10^-1) plus 6 hundredths (10^-2). An essential part of this notation (and a major stumbling block in achieving it) was conceiving of 0 as a number comparable to the other basic digits.

Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations {...,10²,10,1,10^-1,...} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.

Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero, and wherein these four operations obey the usual laws, is called a field.

Addition (+)

Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum.

Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting.

Addition is commutative and associative so the order in which the terms are added does not matter. The identity element of addition (the additive identity) is 0, that is, adding zero to any number will yield that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself will yield the additive identity, 0. For example, the opposite of 7 is (-7), so 7 + (-7) = 0.

Subtraction (−)

Subtraction is essentially the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero.

Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is a − b = a + (−b). When written as a sum, all the properties of addition hold.

Multiplication (× or *)

Multiplication is in essence repeated addition, or the sum of a list of identical numbers. Multiplication finds the product of two numbers, the multiplier and the multiplicand, sometimes both just called factors.

Multiplication, as it is really repeated addition, is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverse is the reciprocal of any number, that is, multiplying the reciprical of any number by the number itself will yield the multiplicative identity, 1.

Division (÷ or /)

Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers and negative one). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × ¹⁄_b. When written as a product, it will obey all the properties of multiplication.

Examples

Addition table

align=center		2	3	4	5	6	7	8	9	align=center	1	2	3	4	5	6	7	8	9	10	align=center	2	3	4	5	6	7	8	9	10	11	align=center	3	4	5	6	7	8	9	10	11	12	align=center	4	5	6	7	8	9	10	11	12	13	align=center	5	6	7	8	9	10	11	12	13	14	align=center	6	7	8	9	10	11	12	13	14	15	align=center	7	8	9	10	11	12	13	14	15	16	align=center	8	9	10	11	12	13	14	15	16	17	align=center	9	10	11	12	13	14	15	16	17	18	align=center	10	11	12	13	14	15	16	17	18	19	20

Multiplication table

align=center		2	3	4	5	6	7	8	9	align=center	1	1	2	3	4	5	6	7	8	9	align=center	2	2	4	6	8	10	12	14	16	18	align=center	3	3	6	9	12	15	18	21	24	27	align=center	4	4	8	12	16	20	24	28	32	36	align=center	5	5	10	15	20	25	30	35	40	45	align=center	6	6	12	18	24	30	36	42	48	54	align=center	7	7	14	21	28	35	42	49	56	63	align=center	8	8	16	24	32	40	48	56	64	72	align=center	9	9	18	27	36	45	54	63	72	81	align=center	10	10	20	30	40	50	60	70	80	90	100

Number theory

The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry. Number theory is also referred to as 'the higher arithmetic', as in the title of H. Davenport's book on the subject.

Arithmetic in education

Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, rational numbers (vulgar fractions), and real numbers (using the decimal place-value system). This study is sometimes known as algorism.

The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960's and 70's, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics http://www.mathematicallycorrect.com/glossary.htm .

Since the introduction of the electronic calculator, which can perform the algorithms far more efficiently than humans, an influential school of educators has argued that mechanical mastery of the standard arithmetic algorithms is no longer necessary. In their view, the first years of school mathematics could be more profitably spent on understanding higher-level ideas about what numbers are used for and relationships among number, quantity, measurement, and so on. However, most research mathematicians still consider mastery of the manual algorithms to be a necessary foundation for the study of algebra and computer science. This controversy was central to the "Math Wars" over California's primary school curriculum in the 1990's, and continues today http://www.education-world.com/a_curr/curr071.shtml .

Footnotes

References

* Cunnington, Susan. The story of arithmetic, a short history of its origin and development. Swan Sonnenschein, London, 1904.
* Dickson, Leonard Eugene. History of the theory of numbers. Three volumes. Reprints: Carnegie Institute of Washington, Washington, 1932. Chelsea, New York, 1952, 1966.
* Fine, Henry Burchard (1858-1928). The number system of algebra treated theoretically and historically. Leach, Shewell & Sanborn, Boston, 1891.
* Karpinski, Louis Charles (1878-1956). The history of arithmetic. Rand McNally, Chicago, 1925. Reprint: Russell & Russell, New York, 1965.
* Ore, Øystein. Number theory and its history. McGraw-Hill, New York, 1948.
* Weil, Andre. Number theory: an approach through history. Birkhauser, Boston, 1984. Reviewed: Math. Rev. 85c:01004.

External links

*What is arithmetic?
*MathWord article about arithmetic

Arithmetic: Encyclopedia BETA

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