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{{short description|Elementary branch of mathematics}}{{for|the song by Brooke Fraser|Arithmetic (song)}}(File:Tables generales aritmetique MG 2108.jpg|thumb|Arithmetic tables for children, Lausanne, 1835)Arithmetic (from the Greek (wikt:en:á¼€ÏÎ¹Î¸Î¼ÏŒÏ‚#Ancient Greek|á¼€ÏÎ¹Î¸Î¼ÏŒÏ‚) arithmos, "number" and (wikt:en:Ï„Î¹ÎºÎ®#Ancient Greek|Ï„Î¹ÎºÎ®) (wikt:en:Ï„Î­Ï‡Î½Î·#Ancient Greek|[Ï„Î­Ï‡Î½Î·)], tikÃ© [tÃ©chne], "art") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on themâ€”addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory.Davenport, Harold, The Higher Arithmetic: An Introduction to the Theory of Numbers (7th ed.), Cambridge University Press, Cambridge, 1999, {{ISBN|0-521-63446-6}}.

Arithmetic operations

{{See also|Algebraic operation}}The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms (Prosthaphaeresis). Arithmetic expressions must be evaluated according to the intended sequence of operations. There are several methods to specify this, eitherâ€”most common, together with infix notationâ€”explicitly using parentheses, and relying on precedence rules, or using a preâ€“ or postfix notation, which uniquely fix the order of execution by themselves. Any set of objects upon which all four arithmetic operations (except division by 0) can be performed, and where these four operations obey the usual laws (including distributivity), is called a field.BOOK, The Oxford Mathematics Study Dictionary, Frank, Tapson, Oxford University Press, 1996, 0-19-914551-2,

If we have two sticks of lengths 2 and 5, then, if we place the sticks one after the other, the length of the stick thus formed is {{math|2 + 5 {{=}} 7}}.

Subtraction (âˆ’)

{{See also|Method of complements}}Subtraction is the inverse operation to addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend: {{math|D {{=}} M - S.}} Resorting to the previously established addition, this is to say that the difference is the number that, when added to the subtrahend, results in the minuend: {{math|D + S {{=}} M.}}For positive arguments {{mvar|M}} and {{mvar|S}} holds:
If the minuend is larger than the subtrahend, the difference {{mvar|D}} is positive. If the minuend is smaller than the subtrahend, the difference {{mvar|D}} is negative.

Multiplication (Ã— or Â· or *)

Multiplication is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, mostly both are simply called factors.Multiplication may be viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number, say x, greater than 1 is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where x was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards 0. (Again, in such a way that 1 goes to the multiplicand.)Another view on multiplication of integer numbers, extendable to rationals, but not very accessible for real numbers, is by considering it as repeated addition. So {{math|3 Ã— 4}} corresponds to either adding {{math|3}} times a {{math|4}}, or {{math|4}} times a {{math|3}}, giving the same result. There are different opinions on the advantageousness of these paradigmata in math education.Multiplication is commutative and associative; further, it is distributive over addition and subtraction. The multiplicative identity is 1, since multiplying any number by 1 yields that same number. The multiplicative inverse for any number except {{math|0}} is the reciprocal of this number, because multiplying the reciprocal of any number by the number itself yields the multiplicative identity {{math|1}}. {{math|0}} is the only number without a multiplicative inverse, and the result of multiplying any number and {{math|0}} is again {{math|0.}} One says that {{math|0}} is not contained in the multiplicative group of the numbers.The product of a and b is written as {{math|a Ã— b}} or {{math|aÂ·b}}. When a or b are expressions not written simply with digits, it is also written by simple juxtaposition: ab. In computer programming languages and software packages in which one can only use characters normally found on a keyboard, it is often written with an asterisk: {{math|a * b.}}Algorithms implementing the operation of multiplication for various representations of numbers are by far more costly and laborious than those for addition. Those accessible for manual computation either rely on breaking down the factors to single place values and apply repeated addition, or employ tables or slide rules, thereby mapping the multiplication to addition and back. These methods are outdated and replaced by mobile devices. Computers utilize diverse sophisticated and highly optimized algorithms to implement multiplication and division for the various number formats supported in their system.

Division (Ã·, or /)

Division is essentially the inverse operation to multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by 0 is undefined. For distinct positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than 1 (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.Division is neither commutative nor associative. So as explained for subtraction, in modern algebra the construction of the division is discarded in favor of constructing the inverse elements with respect to multiplication, as introduced there. That is, division is a multiplication with the dividend and the reciprocal of the divisor as factors, that is {{math|a Ã· b {{=}} a Ã— {{sfrac|1|b}}.}}Within natural numbers there is also a different, but related notion, the Euclidean division, giving two results of "dividing" a natural {{mvar|N}} (numerator) by a natural {{mvar|D}} (denominator), first, a natural {{mvar|Q}} (quotient) and second, a natural {{mvar|R}} (remainder), such that {{math|N {{=}} DÃ—Q + R}} and {{math|R < Q.}}

Compound unit arithmetic{{anchor|Compound Unit Arithmetic}}

CompoundWEB,weblink The Tutor's Companion; or, Complete Practical Arithmetic, Francis, Walkingame, 24â€“39, Webb, Millington & Co, 1860, dead,weblink" title="web.archive.org/web/20150504004020weblink">weblink 2015-05-04, unit arithmetic is the application of arithmetic operations to mixed radix quantities such as feet and inches, gallons and pints, pounds and shillings and pence, and so on. Prior to the use of decimal-based systems of money and units of measure, the use of compound unit arithmetic formed a significant part of commerce and industry.

Basic arithmetic operations

The techniques used for compound unit arithmetic were developed over many centuries and are well-documented in many textbooks in many different languages.BOOK,weblink MÃ©trologie universelle, ancienne et moderne: ou rapport des poids et mesures des empires, royaumes, duchÃ©s et principautÃ©s des quatre parties du monde, French, Universal, ancient and modern metrology: or report of weights and measurements of empires, kingdoms, duchies and principalities of all parts of the world, JFG, Palaiseau, Bordeaux, October 1816, October 30, 2011, BOOK,weblink Allereerste Gronden der Cijferkunst, Jacob de Gelder, 's-Gravenhage and Amsterdam, Dutch, 1824, 163â€“176, de Gebroeders van Cleef, Introduction to Numeracy, March 2, 2011, BOOK, Theoretisch-Praktischer Unterricht im Rechnen fÃ¼r die niederen Classen der Regimentsschulen der KÃ¶nigl. Bayer. Infantrie und Cavalerie, German, Theoretical and practical instruction in arithmetic for the lower classes of the Royal Bavarian Infantry and Cavalry School, MalaisÃ©, Ferdinand, 1842, Munich,weblink 20 March 2012, {{Citation|at=Arithmetick|title=EncyclopÃ¦dia Britannica|volume=Vol I|location=Edinburghstyle="border-bottom-width=0; background:lightblue;"!align="center"|UK pre-decimal currency|4 farthings (f) = 1 penny|12 pennies (d) = 1 shilling|20 shillings (s) = 1 pound (Â£)
title-link=EncyclopÃ¦dia Britannica}} In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions:
• Reduction, in which a compound quantity is reduced to a single quantityâ€”for example, conversion of a distance expressed in yards, feet and inches to one expressed in inches.WEB,weblink The Tutor's Companion; or, Complete Practical Arithmetic, Francis, Walkingame, 43â€“50, Webb, Millington & Co, 1860, dead,weblink" title="web.archive.org/web/20150504004020weblink">weblink 2015-05-04
,
• Expansion, the inverse function to reduction, is the conversion of a quantity that is expressed as a single unit of measure to a compound unit, such as expanding 24 oz to {{nowrap|1 lb, 8 oz}}.
• Normalization is the conversion of a set of compound units to a standard formâ€”for example, rewriting "{{nowrap|1 ft 13 in}}" as "{{nowrap|2 ft 1 in}}".
Knowledge of the relationship between the various units of measure, their multiples and their submultiples forms an essential part of compound unit arithmetic.

Principles of compound unit arithmetic

There are two basic approaches to compound unit arithmetic:
• Reductionâ€“expansion method where all the compound unit variables are reduced to single unit variables, the calculation performed and the result expanded back to compound units. This approach is suited for automated calculations. A typical example is the handling of time by Microsoft Excel where all time intervals are processed internally as days and decimal fractions of a day.
• On-going normalization method in which each unit is treated separately and the problem is continuously normalized as the solution develops. This approach, which is widely described in classical texts, is best suited for manual calculations. An example of the ongoing normalization method as applied to addition is shown below.{|class="infobox bordered" style="font-size: 95%;"
{|class="wikitable" ||style="border: 1px solid #FFFFFF;"(File:MixedUnitAddition.svg|left|thumb)The addition operation is carried out from right to left; in this case, pence are processed first, then shillings followed by pounds. The numbers below the "answer line" are intermediate results.The total in the pence column is 25. Since there are 12 pennies in a shilling, 25 is divided by 12 to give 2 with a remainder of 1. The value "1" is then written to the answer row and the value "2" carried forward to the shillings column. This operation is repeated using the values in the shillings column, with the additional step of adding the value that was carried forward from the pennies column. The intermediate total is divided by 20 as there are 20 shillings in a pound. The pound column is then processed, but as pounds are the largest unit that is being considered, no values are carried forward from the pounds column.For the sake of simplicity, the example chosen did not have farthings.

Operations in practice

(File:Yarloop wkshop gnangarra 14.jpg|thumb|A scale calibrated in imperial units with an associated cost display.)During the 19th and 20th centuries various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pennies and farthings and "Ready Reckoners"â€”books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical bookletBOOK,weblink The Ready Reckoner in miniature containing accurate table from one to the thousand at the various prices from one farthing to one pound., J, Thomson, Montreal, 1824, 25 March 2012, that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound".The cumbersome nature of compound unit arithmetic has been recognized for many yearsâ€”in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth"){{MacTutor|id=Stevin|date=January 2004}} in which he declared the universal introduction of decimal coinage, measures, and weights to be merely a question of time. In the modern era, many conversion programs, such as that included in the Microsoft Windows 7 operating system calculator, display compound units in a reduced decimal format rather than using an expanded format (i.e. "2.5 ft" is displayed rather than {{nowrap|"2 ft 6 in"}}).

Number theory

Until the 19th century, number theory was a synonym of "arithmetic". The addressed problems were directly related to the basic operations and concerned primality, divisibility, and the solution of equations in integers, such as Fermat's last theorem. It appeared that most of these problems, although very elementary to state, are very difficult and may not be solved without very deep mathematics involving concepts and methods from many other branches of mathematics. This led to new branches of number theory such as analytic number theory, algebraic number theory, Diophantine geometry and arithmetic algebraic geometry. Wiles' proof of Fermat's Last Theorem is a typical example of the necessity of sophisticated methods, which go far beyond the classical methods of arithmetic, for solving problems that can be stated in elementary arithmetic.

Arithmetic in education

Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, fractions, and decimals (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 1970s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics.weblink" title="web.archive.org/web/20000519063231weblink">Mathematically Correct: Glossary of TermsAlso, arithmetic was used by Islamic Scholars in order to teach application of the rulings related to Zakat and Irth. This was done in a book entitled The Best of Arithmetic by Abd-al-Fattah-al-Dumyati.WEB, al-Dumyati, Abd-al-Fattah Bin Abd-al-Rahman al-Banna, {{wdl, 3945, |date=1887 |title=The Best of Arithmetic |work=World Digital Library |language=Arabic |accessdate=30 June 2013}}The book begins with the foundations of mathematics and proceeds to its application in the later chapters.

Related topics

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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 (3 volumes), reprints: Carnegie Institute of Washington, Washington, 1932; Chelsea, New York, 1952, 1966
• Euler, Leonhard, weblink" title="web.archive.org/web/20110413234352weblink">Elements of Algebra, Tarquin Press, 2007
• Fine, Henry Burchard (1858â€“1928), The Number System of Algebra Treated Theoretically and Historically, Leach, Shewell & Sanborn, Boston, 1891
• Karpinski, Louis Charles (1878â€“1956), The History of Arithmetic, Rand McNally, Chicago, 1925; reprint: Russell & Russell, New York, 1965
• Ore, Ã˜ystein, Number Theory and Its History, McGrawâ€“Hill, New York, 1948
• Weil, AndrÃ©, Number Theory: An Approach through History, Birkhauser, Boston, 1984; reviewed: Mathematical Reviews 85c:01004

{{Wiktionary}}{{Commons category|Arithmetic}}
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