Arithmeticorarithmetics(from theGreekarithmos, "number") is the oldest[1]and most elementary branch ofmathematics. It consists of the study ofnumbers, especially the properties of the traditionaloperationsbetween themaddition,subtraction,multiplicationanddivision. Arithmetic is an elementary part ofnumber theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along withalgebra,geometry, andanalysis. The termsarithmeticandhigher arithmeticwere used until the beginning of the 20th century as synonyms fornumber theoryand are sometimes still used to refer to a wider part of number theory.[2]

History[edit]The prehistory of arithmetic is limited to a small number of artifacts which may indicate the conception of addition and subtraction, the best-known being theIshango bonefromcentral Africa, dating from somewhere between 20,000 and 18,000BC, although its interpretation is disputed.[3]The earliest written records indicate theEgyptiansandBabyloniansused all theelementary arithmeticoperations as early as 2000BC. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particularnumeral systemstrongly influence the complexity of the methods. The hieroglyphic system forEgyptian numerals, like the laterRoman numerals, descended fromtally marksused for counting. In both cases, this origin resulted in values that used adecimalbase but did not includepositional notation. Complex calculations with Roman numerals required the assistance of acounting boardor theRoman abacusto obtain the results.Early number systems that included positional notation were not decimal, including thesexagesimal(base60) system forBabylonian numeralsand thevigesimal(base20) system that definedMaya numerals. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation.The continuous historical development of modern arithmetic starts with theHellenistic civilizationof ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Prior to the works ofEuclidaround 300BC,Greek studies in mathematicsoverlapped with philosophical and mystical beliefs. For example,Nicomachussummarized the viewpoint of the earlierPythagoreanapproach to numbers, and their relationships to each other, in hisIntroduction to Arithmetic.Greek numeralswere used byArchimedes,Diophantusand others in apositional notationnot very different from ours. Because the ancient Greeks lacked a symbol for zero (until the Hellenistic period), they used three separate sets of symbols. One set for the unit's place, one for the ten's place, and one for the hundred's. Then for the thousand's place they would reuse the symbols for the unit's place, and so on. Their addition algorithm was identical to ours, and their multiplication algorithm was only very slightly different. Their long division algorithm was the same, and the square root algorithm that was once taught in school was known to Archimedes, who may have invented it. He preferred it toHero's methodof successive approximation because, once computed, a digit doesn't change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a fractional part, such as 546.934, they used negative powers of 60 instead of negative powers of 10 for the fractional part 0.934.[4]The ancient Chinese used a similar positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the unit's place, and a second set for the ten's place. For the hundred's place they then reused the symbols for the unit's place, and so on. Their symbols were based on the ancientcounting rods. It is a complicated question to determine exactly when the Chinese started calculating with positional representation, but it was definitely before 400BC.[5]The Bishop of Syria, Severus Sebokht (650AD), "Indians possess a method of calculation that no word can praise enough. Their rational system of mathematics, or of their method of calculation. I mean the system using nine symbols."[6]Leonardo of Pisa (Fibonacci) in 1200AD wrote inLiber Abaci"The method of the Indians (Modus Indoram) surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbolzero".[7]The gradual development ofHinduArabic numeralsindependently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing0. This allowed the system to consistently represent both large and small integers. This approach eventually replaced all other systems. In the early6th century AD,the Indian mathematicianAryabhataincorporated an existing version of this system in his work, and experimented with different notations. In the 7thcentury,Brahmaguptaestablished the use of0 as a separate number and determined the results for multiplication, division, addition and subtraction of zero and all other numbers, except for the result ofdivision by 0. His contemporary, theSyriacbishopSeverus Sebokhtdescribed the excellence of this system as "...valuable methods of calculation which surpass description". The Arabs also learned this new method and called ithesab.

Leibniz'sStepped Reckonerwas the first calculator that could perform all four arithmetic operations.Although theCodex Vigilanusdescribed an early form of Arabic numerals (omitting0) by 976AD, Fibonacci was primarily responsible for spreading their use throughout Europe after the publication of his bookLiber Abaciin 1202. He considered the significance of this "new" representation of numbers, which he styled the "Method of the Indians" (LatinModus Indorum), so fundamental that all related mathematical foundations, including the results ofPythagorasand thealgorismdescribing the methods for performing actual calculations, were "almost a mistake" in comparison.In theMiddle Ages, arithmetic was one of the sevenliberal artstaught in universities.The flourishing ofalgebrain themedievalIslamicworld and inRenaissanceEuropewas an outgrowth of the enormous simplification ofcomputationthroughdecimalnotation.Various types of tools exist to assist in numeric calculations. Examples includeslide rules(for multiplication, division, and trigonometry) andnomographsin addition to the electricalcalculator.Arithmetic operations[edit]See also:Algebraic operationThe basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations ofpercentages,square roots,exponentiation, andlogarithmic functions. Arithmetic is performed according to anorder of operations. Any set of objects upon which all four arithmetic operations (except division by0) can be performed, and where these four operations obey the usual laws, is called afield.[8]Addition (+)[edit]Main article:AdditionAddition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, theaddendsorterms, into a single number, thesumof the numbers (Such as2 + 2 = 4or3 + 5 = 8).Adding more than two numbers can be viewed as repeated addition; this procedure is known assummationand includes ways to add infinitely many numbers in aninfinite series; repeated addition of the number1is the most basic form ofcounting.Addition iscommutativeandassociativeso the order the terms are added in does not matter. Theidentity elementof addition (theadditive identity) is0, that is, adding0 to any number yields that same number. Also, theinverse elementof addition (theadditive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself yields the additive identity,0. For example, the opposite of 7 is 7, so7 + (7) = 0.Addition can be given geometrically as in the following example:If we have two sticks of lengths2and5, then if we place the sticks one after the other, the length of the stick thus formed is2 + 5 = 7.