Arithmetic

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Arithmetic

Arithmetic or arithmetics (from the GreekGreek language

Greek has a documented history of 3,500 years, the longest of any single language within the Indo-European family....
word a???µ?? = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced scienceScience

Science in the broadest sense refers to any system of knowledge attained by verifiable means....
and businessFacts About Business

In economics, business is the social science of managing people to organize and maintain collective productivity toward acco...
calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematicsMathematics

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change....
which records elementary properties of certain operations on numberNumber Summary

A number is an abstract entity that represents a count or measurement....
s. Professional mathematicianMathematician

A mathematician is a person whose primary area of study and research is the field of mathematics....
s sometimes use the term (higher) arithmetic when referring to number theoryNumber theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particu...
, but this should not be confused with elementary arithmeticElementary arithmetic

Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplica...
.

History

The prehistory of arithmetic is limited to a very small number of small artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango boneIshango bone

The Ishango bone is a bone tool, dated to the Upper Paleolithic era, about 18000 to 20000 BC....
from central AfricaDemocratic Republic of the Congo Overview

The Democratic Republic of the Congo, also often referred to as DRC, RDC, DR Congo, Congo or Cong...
, dating from somewhere between 18,000 and 20,000 BC.

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic by 1800 BC, although historians can only guess at the methods utilized to generate the arithmetical results - as shown, for instance, in the clay tablet Plimpton 322Plimpton 322

Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nat...
, which appears to be a list of Pythagorean triplePythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 =...
s, but with no workings to show how the list was originally produced. Likewise, the EgyptianAncient Egypt

Ancient Egypt was a long-lived ancient civilization in north-eastern Africa....
Rhind Mathematical PapyrusRhind Mathematical Papyrus

The Rhind Mathematical Papyrus also designated as: papyrus British Museum 10057, and pBM 10058), is named after Alexander H...
(dating from c. 1650 BC, though evidently a copy of an older text from c. 1850 BC) shows evidence of addition, subtraction, multiplication, and division being used within a unit fractionEgyptian fraction

An Egyptian fraction is the sum of distinct unit fractions whose denominators are positive integers, and all of whose denomi...
system.

NicomachusNicomachus

Nicomachus was born in Gerasa, Roman Syria....
(c. AD6060

EventsBy placeRoman Empire*Boudica sacks London ....
- c. AD120120

Events...
) summarised the philosophical PythagoreanPythagoreanism Summary

Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagorea...
approach to numbers, and their relationships to each other, in his Introduction to ArithmeticIntroduction to Arithmetic

Introduction to Arithmetic was written by Nicomachus almost two thousand years ago, and contains both philosophical prose an...
. At this time, basic arithmetical operations were highly complicated affairs; it was the method known as the "Method of the Indians" (Latin "Modus Indorum") that became the arithmetic that we know today. Indian arithmetic was much simpler than Greek arithmetic due to the simplicity of the Indian number system, which had a zero0 (number)

0 is both a number or, more precisely, a numeral representing a number and a numerical digit....
and place-value notationPositional notation

Positional notation or place-value notation is a numeral system in which each position is related to the next by a con...
. The 7th century7th century

The 7th century is the period from 601 - 700 in accordance with the Julian calendar in the Christian Era....
SyriacSyriac Christianity

Syriac Christianity is a culturally and linguistically distinctive community within Eastern Christianity....
bishop Severus Sebhokt mentioned this method with admiration, stating however that the Method of the Indians was beyond description. The Arabs learned this new method and called it hesab. FibonacciFibonacci

Leonardo of Pisa , also known as Leonardo Pisano, Leonardo Fibonacci, or simply Fibonacci, was an Italian ...
(also known as Leonardo of Pisa) introduced the "Method of the Indians" to Europe in 1202. In his book "Liber AbaciLiber Abaci

Liber Abaci is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci....
", Fibonacci says that, compared with this new method, all other methods had been mistakes. In the Middle AgesMiddle Ages

The Middle Ages formed the middle period in a traditional schematic division of European history into three "ages": the clas...
, arithmetic was one of the seven liberal artsLiberal arts

The term liberal arts has come to mean studies that are intended to provide general knowledge and intellectual skills, rathe...
taught in universities.

Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Arabic numeralsArabic numerals

Arabic numerals, known formally as Hindu-Arabic numerals, and also known as Indian numerals, 'Hindu numerals'...
and decimalDecimal

The decimal numeral system has ten as its base....
place notation for numbers. Arabic numeral based arithmetic was developed by the great Indian mathematicians Aryabhatta, BrahmaguptaBrahmagupta

Brahmagupta was an Indian mathematician and astronomer....
and Bhaskara IBhaskara I

Bhaskara, or Bhaskara I, was a 7th century Indian mathematician, who was apparently the first to write numbers in the ...
. 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 Arabic numerals. Although it is now considered elementary, its simplicitySimplicity

Simplicity is the property, condition, or quality of being simple or un-combined....
is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician ArchimedesArchimedes

Archimedes was an ancient Greek mathematician, physicist, engineer, astronomer, and philosopher born in the seaport colony...
devoted an entire work, The Sand ReckonerThe Sand Reckoner

The Sand Reckoner is probably the most accessible work of Archimedes;...
, to devising a notation for a certain large integer. The flourishing of algebraAlgebra

Algebra is a branch of mathematics concerning the study of structure, relation and quantity....
in the medieval Islamic world and in RenaissanceFacts About Renaissance

In the traditional view, the Renaissance was understood as a historical age in Europe that followed the Middle Ages and ...
EuropeEurope

Europe is one of the seven traditional continents of the Earth....
was an outgrowth of the enormous simplification of computationComputation

Computation is a general term for any type of information processing....
through decimalDecimal

The decimal numeral system has ten as its base....
notation.

Arithmetic operations

The traditional arithmetic operations are additionAddition

Addition is the mathematical operation of increasing one amount by another....
, subtractionSubtraction

Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition....
, multiplicationMultiplication

In mathematics, multiplication is an elementary arithmetic operation....
and divisionDivision (mathematics) Overview

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplicat...
, although more advanced operations (such as manipulations of percentagePercentage

A percentage is a way of expressing numbers as fractions of 100 and is often denoted using the percent sign, "%"....
s, square rootSquare root

In mathematics, a square root of a number x is a number whose square is x....
, exponentiationExponentiation

Exponentiation is a mathematical operation, written a'n, involving two numbers, the base a and the ...
, and logarithmic functionsLogarithm

The logarithm is the mathematical operation that is the inverse of exponentiation ....
) are also sometimes included in this subject. Arithmetic is performed according to an order of operationsOrder of operations

In arithmetic and algebra, certain rules are used for the order in which the operations in expressions are to be evaluated....
. Any set of objects upon which all four operations of arithmetic can be performed (except division by zeroDivision by zero

In mathematics, a division is called a division by zero if the divisor is zero....
), and wherein these four operations obey the usual laws, is called a field.
Addition (+)
Addition is the basic operationOperator

In mathematics, an operator is a function, usually of a special kind depending on the topic....
of arithmetic. In its simplest form, addition combines two numberNumber

A number is an abstract entity that represents a count or measurement....
s, the addends or termsTerm (mathematics)

A term is any value separated by a + or - sign in an expression....
, into a single number, the sum.

Adding more than two numbers can be viewed as repeated addition; this procedure is known as summationSummation

Summation is the addition of a set of numbers; the result is their sum....
and includes ways to add infinitely many numbers in an infinite seriesSeries (mathematics)

In mathematics, a series is often represented as the sum of a sequence of terms....
; repeated addition of the number one1 (number)

1 is a number, numeral, and the name of the glyph representing that number....
is the most basic form of countingCounting

Counting is the mathematical action of repeatedly adding one, usually to find out how many objects there are or to set aside...
.

Addition is commutative and associative so the order in which the terms are added does not matter. The identity elementIdentity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set....
of addition (the additive identityAdditive identity

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when add...
) is 0, that is, adding zero to any number will yield that same number. Also, the inverse elementInverse element

In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, i...
of addition (the additive inverseAdditive inverse

The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero....
) 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.
Addition can be given geometrically as follows.

If a and b are the lengths of two sticks, then if we place the sticks one after the other, the length of the stick
thus formed will be a+b
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 zero0 (number)

0 is both a number or, more precisely, a numeral representing a number and a numerical digit....
.

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 simply called factors.

Multiplication, as it is really repeated addition, is commutative and associative; further it is distributiveDistributivity

In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises th...
over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverseMultiplicative inverse

In mathematics, the reciprocal, or multiplicative inverse, of a number x'' is the number which, when multiplied by '...
is the reciprocal of any number, that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity.
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 zeroDivision by zero

In mathematics, a division is called a division by zero if the divisor is 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). 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 reciprocalMultiplicative inverse

In mathematics, the reciprocal, or multiplicative inverse, of a number x'' is the number which, when multiplied by '...
of the divisor, that is a ÷ b = a × ¹/_b. When written as a product, it will obey all the properties of multiplication.
Examples

Multiplication table
× 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75
4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100
5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125
6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150
7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175
8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200
9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225
10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250
11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 231 242 253 264 275
12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 252 264 276 288 300
13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 273 286 299 312 325
14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280 294 308 322 336 350
15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 375
16 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 336 352 368 384 400
17 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 357 374 391 408 425
18 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450
19 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 399 418 437 456 475
20 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
21 21 42 63 84 105 126 147 168 189 210 231 252 273 294 315 336 357 378 399 420 441 462 483 504 525
22 22 44 66 88 110 132 154 176 198 220 242 264 286 308 330 352 374 396 418 440 462 484 506 528 550
23 23 46 69 92 115 138 161 184 207 230 253 276 299 322 345 368 391 414 437 460 483 506 529 552 575
24 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600
25 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625

Number theory
The term arithmetic is also used to refer to number theoryNumber theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particu...
. This includes the properties of integers related to primalityPrime number

In mathematics, a prime number is a natural number that has exactly two natural number divisors, which are 1 and the prime...
, 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 arithmeticFundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic states that every natural number either is itself a prime number o...
and arithmetic functionArithmetic function

In number theory, an arithmetic function f is a function defined for all positive integers and having values in the com...
s. 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'sHarold Davenport

Harold Davenport was an English mathematician, known for his extensive work in number theory....
book on the subject.
Arithmetic in education
Primary educationPrimary education

Primary or elementary education consists of the first years of formal, structured education that occurs during childho...
in mathematics often places a strong focus on algorithms for the arithmetic of natural numberNatural number

In mathematics, a natural number is either a positive integer or a non-negative integer ....
s, integerInteger

The integers consist of the positive natural numbers , their negatives and the number zero....
s, rational numberRational number

In mathematics, a rational number is a ratio or quotient of two integers, usually written as the vulgar fraction a''/b'...
s, and real numberReal number

In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers....
s (using the decimalDecimal

The decimal numeral system has ten as its base....
place-value system). This study is sometimes known as algorismAlgorism

Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in...
.

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 MathNew math Summary

New math is a term referring to a brief dramatic change in the way mathematics was taught in American grade schools during t...
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 calculatorCalculator

A calculator is a device for performing calculations....
, 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 WarsMath wars

Math wars is the debate over modern mathematics education, textbooks and curricula in the United States that was triggered b...
" over California's primary school curriculum in the 1990s, and continues today.

Many mathematics texts for K-12 instruction were developed, funded by grants from the United States National Science FoundationNational Science Foundation

The National Science Foundation is an independent United States government agency that supports fundamental research and ed...
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 TERCTERC

TERC may refer to:*Telomerase RNA component, a human gene....
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 arithmetic, teachers are instructed to instead guide students to invent their own (some critics claim inefficient) methods, instead using such techniques as skip countingSkip counting

Skip counting is a mathematics technique taught as a kind of multiplication in standards-based mathematics textbooks such as...
, 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 mathematicsTraditional mathematics

Traditional mathematics is the term used for the style of mathematics instruction used before the adoption of standards-base...
texts and discarded such reform-based approaches as misguided failures.
See also

Lists
List of basic arithmetic topicsList of basic arithmetic topics Summary

Basic topics in arithmetic include:...
List of mathematics topics

Related topics

External links

(historical)

an early western work on arithmetic at