Arithmetic tables for children, Lausanne, 1835

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^[1] when referring to number theory, but this should not be confused with elementary arithmetic. Image File history File linksMetadata Size of this preview: 399 × 599 pixelsFull resolution (2691 × 4038 pixel, file size: 1,015 KB, MIME type: image/jpeg) Work by Rama File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Arithmetic Metadata... Image File history File linksMetadata Size of this preview: 399 × 599 pixelsFull resolution (2691 × 4038 pixel, file size: 1,015 KB, MIME type: image/jpeg) Work by Rama File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Arithmetic Metadata... Part of a scientific laboratory at the University of Cologne. ... Wall Street, Manhattan is the location of the New York Stock Exchange and is often used as a symbol for the world of business. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A number is an abstract idea used in counting and measuring. ... Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ...

History

See also: Introduction to Arithmetic

It is the method known as the "Method of the Indians" or in Latin "Modus Indorum" that has become our arithmetic today. Prior to this, basic arithmetic operations were highly complicated affairs. Seventh century Syriac Bishop Severus Sebhokt mentioned this method and stated that the method of the Indians is beyond description. Indian arithmetic was much simpler than the Greek arithmetic simply due to the simplicity of the Indian number system which had a zero and place value notation. Arabs learned this new method and called it "Hesab" or "Hindu Science". Fibonacci or Leonardo of Pisa is one of the first European mathematicians who introduced the "Method of the Indians" to Europe in 1202. In his famous book "Liber Abaci" Fibonacci says that compared to this new method all other methods were mistakes. Introduction to Arithmetic was written by Nicomachus almost two thousand years ago, and contains both philosophical prose and very basic mathematical ideas. ... Leonardo of Pisa (1170s or 1180s – 1250), also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some the most talented mathematician of the Middle Ages. ...

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. The Ishango bone is a tally stick, made of bone, which contains sequences of prime numbers, and some series of multiples. ... A world map showing the continent of Africa Africa is the worlds second-largest and second most-populous continent, after Asia. ...

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic circa 1850 BC, although historians can only infer the methods utilized to generate the arithmetical results (see Plimpton 322). Likewise, definitive addition, subtraction, multiplication, and division facts are used within the unit fraction system, which can be found in the Rhind Mathematical Papyrus dating from Ancient Egypt circa 1650 BC, copied from 1850 BC (Mathematathe septem liberales artes (seven liberal arts). Babylonia was an ancient state in Iraq), combining the territories of Sumer and Akkad. ... Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ...

Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Hindu-Arabic numerals and decimal place notation for numbers. Hindu- Arabic numeral based arithmetic was developed by great Indian mathematicians Aryabhatta, Brahmagupta and Bhaskara. Aryabhatta tried different place value notations and Brahmagupta added zero to the Indian number system. Brahmagupta developed modern multiplication, division, addition and subtraction based on Hindu-Arabic numerals. 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. Numerals sans-serif Arabic numerals, known formally as Hindu-Arabic numerals, and also as Indian numerals, Hindu numerals, Western Arabic numerals, European numerals, or Western numerals, are the most common symbolic representation of numbers around the world. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... Archimedes of Syracuse (Greek: c. ... The Sand Reckoner is probably the most accessible work of Archimedes, in some sense, it is the first research-expository paper. ...

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. Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus...

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. 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...

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. 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... The percent sign. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. ... This article or section does not cite its references or sources. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Addition (+)

Main article: 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. 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... A number is an abstract idea used in counting and measuring. ... In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ...

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. For evaluation of sums in closed form see evaluating sums. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... Look up one in Wiktionary, the free dictionary. ... Counting is the mathematical action of repeatedly adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers...

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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... The additive identity of a number n is the number which, when added to n will yield n. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...

Subtraction (−)

Main article: 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. 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... For other uses, see zero or 0. ...

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 ·)

Main article: Multiplication

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 simply called factors. In mathematics, multiplication is an elementary arithmetic operation. ...

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. In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... One redirects here. ... The reciprocal function: y = 1/x. ... The reciprocal function: y = 1/x. ...

Division (÷ or /)

Main article: Division (mathematics)

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. In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... This article or section does not cite its references or sources. ...

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. The reciprocal function: y = 1/x. ...

Examples

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. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... In number theory and computability theory, subfields of mathematics, a number-theoretic function is any function whose domain is the set of natural numbers. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Harold Davenport (30 October 1907 - 9 June 1969) was an English mathematician, known for his extensive work in number theory. ...

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. This article or section does not cite any references or sources. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... 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...

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.^[2] New math is a term referring to a brief dramatic change in the way mathematics was taught in American grade schools during the 1960s. ...

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.^[3] A calculator is a device for performing calculations. ...

Many mathematics texts for K-12 instruction were developed, funded by grants from the United States National Science Foundation based on standards created by the NCTM and given high ratings by United States Department of Education, though condemned by many mathematicians. Some widely adopted texts such as TERC were based on the spirit of research papers which found that instruction of basic arithmetic was harmful to mathematical understanding. Rather than teaching any traditional method of arithemtic, teachers are instructed to instead guide students to invent their own (some critics claim inefficient) methods, instead using such techniques as skip counting, and the heavy use of manipulatives, scissors and paste, and even singing rather than multiplication tables or long division. Although such texts were designed to be a complete curricula, in the face of intense protest and criticism, many districts have chosen to circumvent the intent of such radical approaches by supplementing with traditional texts. Other districts have since adopted traditional mathematics texts and discarded such reform-based approaches as misguided failures. The logo of the National Science Foundation The National Science Foundation (NSF) is an independent United States government agency that supports fundamental research and education in all the non-medical fields of science and engineering. ... The National Council of Teachers of Mathematics (NCTM) was founded in 1920. ... Investigations in Number, Data, and Space is a complete K-5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts. ... Skip counting is a mathematics technique taught in place of formal multiplication in standards-based mathematics textbooks such as TERC. Another similar method is coloring in squares in a 100s chart to show multiplication patterns. ... Traditional mathematics is the term used for the style of mathematics instruction used for a period in the 20th century before the appearance of reform mathematics based on NCTM standards, so it is best defined by contrast with the alternatives. ...

See also

Lists

• List of basic arithmetic topics
• List of mathematics topics

Arithmetic is the oldest and simplest branch of mathematics, used by almost everyone. ... These list of mathematical topics pages collect pointers to all articles related to mathematics. ...

Related topics

Addition of natural numbers is the most basic arithmetic operation. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... In mathematics, associativity is a property that a binary operation can have. ... A map or binary operation from a set to a set is said to be commutative if, (A common example in school-math is the + function: , thus the + function is commutative) Otherwise, the operation is noncommutative. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ... Arithmetic in a finite field is different from standard integer arithmetic. ... A number line, invented by John Wallis, is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a line. ... This is a list of important publications in mathematics, organized by field. ... Arithmetic coding is a method for lossless data compression. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ...

Footnotes

References

Louis Charles Karpinski (1878–1956) was an American mathematician born in Rochester, N. Y. and educated at Cornell and in Europe at Strassburg. ...

External links

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Arithmetic

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