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Arithmetic

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Arithmetic or arithmetics (from the Greek word αριθμός = number) is the
   oldest and most elementary 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 arithmetic 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 small
   artifacts indicating a clear conception of addition and subtraction,
   the best-known being the Ishango Bone from Africa, dating from 18,000
   BC.

   It is clear that the Babylonians had solid knowledge of almost all
   aspects of elementary arithmetic circa 1850 BC, 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 BC.

   In the Pythagorean school, in the second half of the 6th century BC,
   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 (10^2), plus 0 tens (10^1), plus 7
   units (10^0), 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 {...,10^2,10,1,10^-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 reciprocal 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


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

                                   Multiplication table


                                   ×  1  2  3  4  5  6  7  8  9  10
                                   1  1  2  3  4  5  6  7  8  9  10
                                   2  2  4  6  8  10 12 14 16 18 20
                                   3  3  6  9  12 15 18 21 24 27 30
                                   4  4  8  12 16 20 24 28 32 36 40
                                   5  5  10 15 20 25 30 35 40 45 50
                                   6  6  12 18 24 30 36 42 48 54 60
                                   7  7  14 21 28 35 42 49 56 63 70
                                   8  8  16 24 32 40 48 56 64 72 80
                                   9  9  18 27 36 45 54 63 72 81 90
                                   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 1960s and '70s,
   which attempted to teach arithmetic in the spirit of axiomatic
   development from set theory, an echo of the prevailing trend in higher
   mathematics .

   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 1990s, and continues
   today .

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