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 day-to-day counting to advanced science and business calculations. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding A business (also called firm or an enterprise) is a legally recognized organizational entity designed to provide goods and/or services to In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. Professional mathematicians sometimes use the term (higher) arithmetic^[1] when referring to number theory, but this should not be confused with elementary arithmetic. 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 Elementary arithmetic is the most basic kind of Mathematics: it concerns the operations of Addition, Subtraction, Multiplication, and division

Contents 1 History 2 Decimal arithmetic 3 Arithmetic operations 3.1 Addition (+) 3.2 Subtraction (−) 3.3 Multiplication (×, ·, or *) 3.4 Division (÷ or /) 3.5 Examples 3.5.1 Multiplication table 4 Number theory 5 Arithmetic in education 6 See also 6.1 Lists 6.2 Related topics 7 Footnotes 8 References 9 External links

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 bone from central Africa, dating from somewhere between 18,000 and 20,000 BC. The Ishango bone is a Bone tool, dated to the Upper Paleolithic era about 18000 to 20000 BC The Democratic Republic of the Congo (République démocratique du Congo often referred to as DR Congo, DRC or RDC, and formerly known or referred to

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 322, which appears to be a list of Pythagorean triples, but with no workings to show how the list was originally produced. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital Of the approximately half million Babylonian Clay tablets excavated since the beginning of the 19th century several thousand are of a mathematical nature A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  Likewise, the Egyptian Rhind Mathematical Papyrus (dating from c. Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish 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 fraction system. An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16}

Nicomachus (c. Nicomachus (Νικόμαχος (c 60 &ndash c 120 was an important mathematician in the ancient world and is best known for his works Introduction to Arithmetic AD60 - c. Year 60 was a Leap year starting on Tuesday (link will display the full calendar of the Julian calendar. AD120) summarised the philosophical Pythagorean approach to numbers, and their relationships to each other, in his Introduction to Arithmetic. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced Introduction to Arithmetic was written by Nicomachus almost two thousand years ago and contains both philosophical prose and very basic mathematical ideas 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 zero and place-value notation. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a The 7th century Syriac bishop Severus Sebhokt mentioned this method with admiration, stating however that the Method of the Indians was beyond description. The 7th century is the period from 601 to 700 in accordance with the Julian calendar in the Christian / Common Era. Syriac Christianity is a culturally and linguistically distinctive community within Eastern Christianity. The Arabs learned this new method and called it hesab. Fibonacci (also known as Leonardo of Pisa) introduced the "Method of the Indians" to Europe in 1202. Leonardo of Pisa (c 1170 – c 1250 also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or most commonly simply Fibonacci In his book "Liber Abaci", Fibonacci says that, compared with this new method, all other methods had been mistakes. Liber Abaci (1202 also spelled as Liber Abbaci) is an historic book on Arithmetic by Leonardo of Pisa known later by his nickname Fibonacci In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities. The term liberal arts refers to a particular type of educational Curriculum broadly defined as a Classical education.

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. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system The decimal ( base ten or occasionally denary) Numeral system has ten as its base. Arabic numeral based arithmetic was developed by the great Indian mathematicians Aryabhatta, Brahmagupta and Bhāskara I. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Bhāskara (commonly called Bhāskara I to avoid confusion with the 12th century mathematician Bhāskara II) (c 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 simplicity is the culmination of thousands of years of mathematical development. Simplicity is the property condition or quality of being simple or un-combined By contrast, the ancient mathematician Archimedes devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer The Sand Reckoner ( Greek: Ψαμμίτης Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number 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. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. For other meanings including people named 'Islam' see Islam (disambiguation. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere Computation is a general term for any type of Information processing. The decimal ( base ten or occasionally denary) Numeral system has ten as its base.

Decimal arithmetic

Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. . . ,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 zero 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. Algorism is the technique of performing basic Arithmetic by writing numbers in Place value form and applying a set of memorized rules and facts to the digits 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.

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. Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Mathematics, a percentage is a way of expressing a number as a Fraction of 100 ( per cent meaning "per hundred" In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce Arithmetic is performed according to an order of operations. In Algebra and Computer programming, when a number or expression is both preceded and followed by a Binary operation, a rule is required for which operation 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. In In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Addition (+)

Main article: Addition

Addition is the basic operation of arithmetic. Addition is the mathematical process of putting things together In Mathematics, an operator is a function which operates on (or modifies another function In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. The word term is from the Latin terminus "boundary line limit" from the Indo-European root ter- "peg post boundary"

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. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity 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

Addition is commutative and associative so the order in which the terms are added does not matter. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have The identity element of addition (the additive identity) is 0, that is, adding zero to any number will yield that same number. 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 In Mathematics the additive identity of a set which is equipped with the operation of Addition is an element which when added to 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. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero. 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 (−)

Main article: Subtraction

Subtraction is essentially the opposite of addition. Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract 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 *)

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.

Multiplication, as it is really repeated addition, is commutative and associative; further it is distributive over addition and subtraction. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity 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. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which

Division (÷ or /)

Main article: Division (mathematics)

Division is essentially the opposite of multiplication. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. In 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 reciprocal of the divisor, that is a ÷ b = a × ¹⁄_b. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which 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

Main article: Number theory

The term arithmetic is also used to refer to number theory. 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 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 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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Number theory an arithmetic function or arithmetical function is a Function defined on the set of Natural numbers (i 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. 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). Primary education is the first stage of Compulsory education. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object In Mathematics, the real numbers may be described informally in several different ways The decimal ( base ten or occasionally denary) Numeral system has ten as its base. This study is sometimes known as algorism. Algorism is the technique of performing basic Arithmetic by writing numbers in Place value form and applying a set of memorized rules and facts to the digits

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. New Math was a brief dramatic change in the way Mathematics was taught in American Grade schools during the 1960s The name is commonly given ^[2]

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. A calculator is device for performing mathematical calculations distinguished from a Computer by having a limited problem solving ability and an interface optimized for interactive 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. Math wars is the debate over modern Mathematics education, textbooks and curricula in the United States that was triggered by the publication in 1989 of the Curriculum ^[3]

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. The National Science Foundation (NSF is a United States Government agency that supports fundamental Research and Education in all the non-medical The National Council of Teachers of Mathematics (NCTM was founded in 1920. 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 arithmetic, 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. Skip counting is a mathematics technique taught as a kind of multiplication in Standards-based mathematics textbooks such as TERC. 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. Traditional mathematics (sometimes classical math education) is a term used to describe the predominant methods of Mathematics education in the United States

See also

Lists

• List of basic arithmetic topics
• List of mathematics topics

Related topics

Footnotes

References

External links

• What is arithmetic?
• MathWorld article about arithmetic
• Interactive Arithmetic Lessons and Practice
• Talking Math Game for kids
• The New Student's Reference Work/Arithmetic (historical)
• Arithmetic Game
• Math Games for kids and adults
• Maximus Planudes' the Great Calculation an early western work on arithmetic at Convergence

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