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Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being a ...
,
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
,
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
, and extraction of roots. In the 19th century, Italian mathematician
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The stan ...
formalized arithmetic with his
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
, which are highly important to the field of
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal s ...
today.


History

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 the
Ishango bone The Ishango bone, discovered at the "Fisherman Settlement" of Ishango in the Democratic Republic of Congo, is a bone tool and possible mathematical device that dates to the Upper Paleolithic era. The curved bone is dark brown in color, about 10 c ...
from
central Africa Central Africa is a subregion of the African continent comprising various countries according to different definitions. Angola, Burundi, the Central African Republic, Chad, the Democratic Republic of the Congo, the Republic of the Cong ...
, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed. The earliest written records indicate the
Egyptians Egyptians ( arz, المَصرِيُون, translit=al-Maṣriyyūn, ; arz, المَصرِيِين, translit=al-Maṣriyyīn, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group native to the Nile, Nile Valley in Egypt. Egyptian ...
and
Babylonians Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state ...
used all the
elementary arithmetic The operators in elementary arithmetic are addition, subtraction, multiplication, and division. The operators can be applied on both real numbers and imaginary numbers. Each kind of number is represented on a number line designated to the type. ...
operations: addition, subtraction, multiplication, and division, as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...
strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later
Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ...
, descended from
tally marks Tally marks, also called hash marks, are a unary numeral system ( arguably). They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate ...
used for counting. In both cases, this origin resulted in values that used a
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
base, but did not include
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
. Complex calculations with Roman numerals required the assistance of a
counting board The counting board is the precursor of the abacus, and the earliest known form of a counting device (excluding fingers and other very simple methods). Counting boards were made of stone or wood, and the counting was done on the board with beads, ...
(or the
Roman abacus The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of earlier abacuses like those that were used by the Greeks and Babylonians. Origin The Roman abacus was the first portable calculating device for e ...
) to obtain the results. Early number systems that included positional notation were not decimal; these include the
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form� ...
(base 60) system for
Babylonian numerals Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were fa ...
, and the
vigesimal A vigesimal () or base-20 (base-score) numeral system is based on 20 (number), twenty (in the same way in which the decimal, decimal numeral system is based on 10 (number), ten). ''wikt:vigesimal#English, Vigesimal'' is derived from the Latin ad ...
(base 20) system that defined
Maya numerals The Mayan numeral system was the system to represent numbers and calendar dates in the Maya civilization. It was a vigesimal (base-20) positional numeral system. The numerals are made up of three symbols; zero (a shell), one (a dot) and fi ...
. Because of the 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 the
Hellenistic period In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
of ancient Greece; it originated much later than the Babylonian and Egyptian examples. Prior to the works of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs.
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and '' Manual of Harmonics'' in Greek. He was born i ...
is an example of this viewpoint, using the earlier Pythagorean approach to numbers and their relationships to each other in his work ''
Introduction to Arithmetic The book ''Introduction to Arithmetic'' ( grc-gre, Ἀριθμητικὴ εἰσαγωγή, ''Arithmetike eisagoge'') is the only extant work on mathematics by Nicomachus (60–120 AD). Summary The work contains both philosophical prose and ...
''.
Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those ...
were used by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
,
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
and others in a
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
not very different from the modern notation. The ancient Greeks lacked a symbol for zero until the Hellenistic period, and they used three separate sets of symbols as digits: one set for the units place, one for the tens place, and one for the hundreds. For the thousands place, they would reuse the symbols for the units place, and so on. Their addition algorithm was identical to the modern method, and their multiplication algorithm was only slightly different. Their long division algorithm was the same, and the digit-by-digit square root algorithm, popularly used as recently as the 20th century, was known to Archimedes (who may have invented it). He preferred it to Hero's method of successive approximation because, once computed, a digit does not 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. The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty and continuing through the Tang Dynasty, from basic numbers to advanced algebra. The ancient Chinese used a positional notation similar to that of the Greeks. Since they also lacked a symbol for
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
, they had one set of symbols for the units place, and a second set for the tens place. For the hundreds place, they then reused the symbols for the units place, and so on. Their symbols were based on the ancient
counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written ...
. The exact time where the Chinese started calculating with positional representation is unknown, though it is known that the adoption started before 400 BC. The ancient Chinese were the first to meaningfully discover, understand, and apply negative numbers. This is explained in the ''
Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'' (''Jiuzhang Suanshu''), which was written by
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state o ...
dated back to 2nd century BC. The gradual development of the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syst ...
independently 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 representing 0. This allowed the system to consistently represent both large and small integers—an approach which eventually replaced all other systems. In the early the Indian mathematician
Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (whic ...
incorporated an existing version of this system in his work, and experimented with different notations. In the 7th century,
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
established the use of 0 as a separate number, and determined the results for multiplication, division, addition and subtraction of zero and all other numbers—except for the result of
division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
. His contemporary, the
Syriac Syriac may refer to: *Syriac language, an ancient dialect of Middle Aramaic *Sureth, one of the modern dialects of Syriac spoken in the Nineveh Plains region * Syriac alphabet ** Syriac (Unicode block) ** Syriac Supplement * Neo-Aramaic languages ...
bishop
Severus Sebokht Severus Sebokht ( syc, ܣܘܪܘܣ ܣܝܒܘܟܬ), also Seboukt of Nisibis, was a Syrian scholar and bishop who was born in Nisibis, Syria in 575 and died in 667. Although little is known about his early life, he was one of the leading figures in S ...
(650 AD) said, "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." The Arabs also learned this new method and called it ''hesab''. Although the
Codex Vigilanus The ''Codex Vigilanus'' or ''Codex Albeldensis'' (Spanish: ''Códice Vigilano'' or ''Albeldense'') is an illuminated compilation of various historical documents accounting for a period extending from antiquity to the 10th century in Hispania ...
described an early form of Arabic numerals (omitting 0) by 976 AD, Leonardo of Pisa (
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
) was primarily responsible for spreading their use throughout Europe after the publication of his book ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describ ...
'' in 1202. He wrote, "The method of the Indians (Latin ''Modus Indorum'') surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbol
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
". In the Middle Ages, arithmetic was one of the seven
liberal arts Liberal arts education (from Latin "free" and "art or principled practice") is the traditional academic course in Western higher education. ''Liberal arts'' takes the term ''art'' in the sense of a learned skill rather than specifically the ...
taught in universities. The flourishing of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
in the
medieval In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
Islamic Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God (or '' Allah'') as it was revealed to Muhammad, the ma ...
world, and also in
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
Europe, was an outgrowth of the enormous simplification of
computation Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm). Mechanical or electronic devices (or, historically, people) that perform computations are known as ''computers''. An esp ...
through
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
notation. Various types of tools have been invented and widely used to assist in numeric calculations. Before Renaissance, they were various types of
abaci The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hi ...
. More recent examples include
slide rule The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which i ...
s,
nomogram A nomogram (from Greek , "law" and , "line"), also called a nomograph, alignment chart, or abac, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. ...
s and
mechanical calculator A mechanical calculator, or calculating machine, is a mechanical device used to perform the basic operations of arithmetic automatically, or (historically) a simulation such as an analog computer or a slide rule. Most mechanical calculators we ...
s, such as Pascal's calculator. At present, they have been supplanted by electronic
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s and
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...
s.


Arithmetic operations

The basic arithmetic operations are addition, subtraction, multiplication and division, although arithmetic also includes more advanced operations, such as manipulations of
percentage In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also u ...
s,
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
s,
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
,
logarithmic function In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s, 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, either—most common, together with
infix notation Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—" infixed operators"—such as the plus sign in . Usage Binary relations are ...
—explicitly using parentheses and relying on precedence rules, or using a
prefix A prefix is an affix which is placed before the stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy''. Particula ...
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 zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
) can be performed, and where these four operations obey the usual laws (including distributivity), is called a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
.


Addition

Addition, denoted by the symbol +, is the most basic operation of arithmetic. In its simple form, addition combines two numbers, the ''addends'' or ''terms'', into a single number, the ''sum'' of the numbers (such as or ). Adding finitely many numbers can be viewed as repeated simple addition; this procedure is known as
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matr ...
, a term also used to denote the definition for "adding infinitely many numbers" in an
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mat ...
. Repeated addition of the number  1 is the most basic form of
counting Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every e ...
; the result of adding is usually called the
successor Successor may refer to: * An entity that comes after another (see Succession (disambiguation)) Film and TV * ''The Successor'' (film), a 1996 film including Laura Girling * ''The Successor'' (TV program), a 2007 Israeli television program Musi ...
of the original number. Addition is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
and
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
, so the order in which finitely many terms are added does not matter. The number has the property that, when added to any number, it yields that same number; so, it is the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
of addition, or the additive identity. For every number , there is a number denoted , called the '' opposite'' of , such that and . So, the opposite of is the inverse of with respect to addition, or the
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
of . For example, the opposite of is , since . Addition can also be interpreted geometrically, as in the following example. If we have two sticks of lengths ''2'' and ''5'', then, if the sticks are aligned one after the other, the length of the combined stick becomes ''7'', since .


Subtraction

Subtraction, denoted by the symbol -, is the inverse operation to addition. Subtraction finds the ''difference'' between two numbers, the ''minuend'' minus the ''subtrahend'': 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: For positive arguments and holds: :If the minuend is larger than the subtrahend, the difference is positive. :If the minuend is smaller than the subtrahend, the difference is negative. In any case, if minuend and subtrahend are equal, the difference Subtraction is neither
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
nor
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
. For that reason, the construction of this inverse operation in modern algebra is often discarded in favor of introducing the concept of inverse elements (as sketched under ), where subtraction is regarded as adding the additive inverse of the subtrahend to the minuend, that is, . The immediate price of discarding the binary operation of subtraction is the introduction of the (trivial)
unary operation In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
, delivering the additive inverse for any given number, and losing the immediate access to the notion of difference, which is potentially misleading when negative arguments are involved. For any representation of numbers, there are methods for calculating results, some of which are particularly advantageous in exploiting procedures, existing for one operation, by small alterations also for others. For example, digital computers can reuse existing adding-circuitry and save additional circuits for implementing a subtraction, by employing the method of
two's complement Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
for representing the additive inverses, which is extremely easy to implement in hardware (
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
). The trade-off is the halving of the number range for a fixed word length. A formerly widespread method to achieve a correct change amount, knowing the due and given amounts, is the ''counting up method'', which does not explicitly generate the value of the difference. Suppose an amount ''P'' is given in order to pay the required amount ''Q'', with ''P'' greater than ''Q''. Rather than explicitly performing the subtraction ''P'' − ''Q'' = ''C'' and counting out that amount ''C'' in change, money is counted out starting with the successor of ''Q'', and continuing in the steps of the currency, until ''P'' is reached. Although the amount counted out must equal the result of the subtraction ''P'' − ''Q'', the subtraction was never really done and the value of ''P'' − ''Q'' is not supplied by this method.


Multiplication

Multiplication, denoted by the symbols \times or \cdot, 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 called ''factors''. Multiplication may be viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number greater than 1, say ''x'', 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, 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. For example. corresponds to either adding times a , or times a , 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 In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
is 1, since multiplying any number by 1 yields that same number. The
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
for any number except  is the reciprocal of this number, because multiplying the reciprocal of any number by the number itself yields the multiplicative identity .  is the only number without a multiplicative inverse, and the result of multiplying any number and is again One says that is not contained in the multiplicative
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of the numbers. The product of ''a'' and ''b'' is written as or . It can 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: 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 applying repeated addition, or on employing
tables Table may refer to: * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (landform), a flat area of land * Table (information), a data arrangement with rows and columns * Table (database), how the table da ...
or
slide rules The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which i ...
, thereby mapping multiplication to addition and vice versa. These methods are outdated and are gradually replaced by mobile devices. Computers use diverse sophisticated and highly optimized algorithms, to implement multiplication and division for the various number formats supported in their system.


Division

Division, denoted by the symbols \div or /, is essentially the inverse operation to multiplication. Division finds the ''quotient'' of two numbers, the ''dividend'' divided by the ''divisor''. Under common rules, dividend divided by zero 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 or equal to 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 in , the construction of the division in modern algebra is discarded in favor of constructing the inverse elements with respect to multiplication, as introduced in . Hence division is the multiplication of the dividend with the reciprocal of the divisor as factors, that is, Within the natural numbers, there is also a different but related notion called
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
, which outputs two numbers after "dividing" a natural (numerator) by a natural (denominator): first a natural (quotient), and second a natural (remainder) such that and In some contexts, including computer programming and advanced arithmetic, division is extended with another output for the remainder. This is often treated as a separate operation, the
Modulo operation In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is th ...
, denoted by the symbol % or the word mod, though sometimes a second output for one "divmod" operation. In either case,
Modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
has a variety of use cases. Different implementations of division (floored, truncated, Euclidean, etc.) correspond with different implementations of modulus.


Fundamental theorem of arithmetic

The fundamental theorem of arithmetic states that any integer greater than 1 has a unique prime factorization (a representation of a number as the product of prime factors), excluding the order of the factors. For example, 252 only has one prime factorization: :252 = 2 × 3 × 7 Euclid's ''Elements'' first introduced this theorem, and gave a partial proof (which is called
Euclid's lemma In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as we ...
). The fundamental theorem of arithmetic was first proven by
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
. The fundamental theorem of arithmetic is one of the reasons why 1 is not considered a prime number. Other reasons include the
sieve of Eratosthenes In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime ...
, and the definition of a prime number itself (a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.).


Decimal arithmetic

refers exclusively, in common use, to the written
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...
employing
arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
as the digits for a
radix In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
10 ("decimal")
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
; however, any
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...
based on powers of 10, e.g.,
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
,
Cyrillic The Cyrillic script ( ), Slavonic script or the Slavic script, is a writing system used for various languages across Eurasia. It is the designated national script in various Slavic, Turkic, Mongolic, Uralic, Caucasian and Iranic-speaking cou ...
,
Roman Roman or Romans most often refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people, the people of ancient Rome *''Epistle to the Romans'', shortened to ''Romans'', a letter ...
, or
Chinese numerals Chinese numerals are words and characters used to denote numbers in Chinese. Today, speakers of Chinese use three written numeral systems: the system of Arabic numerals used worldwide, and two indigenous systems. The more familiar indigenous ...
may conceptually be described as "decimal notation" or "decimal representation". Modern methods for four fundamental operations (addition, subtraction, multiplication and division) were first devised by
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
of India. This was known during medieval Europe as "Modus Indorum" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s using the same symbol for the different
orders of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic di ...
(e.g., the "ones place", "tens place", "hundreds place") and, with a
radix point A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The cho ...
, using those same symbols to represent
fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
(e.g., the "tenths place", "hundredths place"). 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). The concept of 0 as a number comparable to the other basic digits is essential to this notation, as is the concept of 0's use as a placeholder, and as is the definition of multiplication and addition with 0. The use of 0 as a placeholder and, therefore, the use of a positional notation is first attested to in the
Jain Jainism ( ), also known as Jain Dharma, is an Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current time cycle being ...
text from
India India, officially the Republic of India ( Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the ...
entitled the '' Lokavibhâga'', dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
using the Hindu–Arabic numeral system.
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. One who practices algorism is known as an algorist. This positional notation system ...
comprises all of the rules for performing arithmetic computations using this type of written numeral. For example, addition produces the sum of two arbitrary numbers. The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left. An addition table with ten rows and ten columns displays all possible values for each sum. If an individual sum exceeds the value 9, the result is represented with two digits. The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one (if not zero). This adjustment is termed a ''carry'' of the value 1. The process for multiplying two arbitrary numbers is similar to the process for addition. A multiplication table with ten rows and ten columns lists the results for each pair of digits. If an individual product of a pair of digits exceeds 9, the ''carry'' adjustment increases the result of any subsequent multiplication from digits to the left by a value equal to the second (leftmost) digit, which is any value from (). Additional steps define the final result. Similar techniques exist for subtraction and division. The creation of a correct process for multiplication relies on the relationship between values of adjacent digits. The value for any single digit in a numeral depends on its position. Also, each position to the left represents a value ten times larger than the position to the right. In mathematical terms, the
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
for the
radix In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
(base) of 10 increases by 1 (to the left) or decreases by 1 (to the right). Therefore, the value for any arbitrary digit is multiplied by a value of the form 10''n'' with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
 ''n''. The list of values corresponding to all possible positions for a single digit is written Repeated multiplication of any value in this list by 10 produces another value in the list. In mathematical terminology, this characteristic is defined as closure, and the previous list is described as . It is the basis for correctly finding the results of multiplication using the previous technique. This outcome is one example of the uses of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
.


Compound unit arithmetic

Compound unit arithmetic is the application of arithmetic operations to
mixed radix Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a ...
quantities such as feet and inches; gallons and pints; pounds, shillings and pence; and so on. Before decimal-based systems of money and units of measure, compound unit arithmetic was widely used in commerce and industry.


Basic arithmetic operations

The techniques used in compound unit arithmetic were developed over many centuries and are well documented in many textbooks in many different languages. In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions: * , in which a compound quantity is reduced to a single quantity—for example, conversion of a distance expressed in yards, feet and inches to one expressed in inches. * , the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
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 . * is the conversion of a set of compound units to a standard form—for example, rewriting "" as "". 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: * 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 Microsoft Excel is a spreadsheet developed by Microsoft for Windows, macOS, Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for Ap ...
where all time intervals are processed internally as days and decimal fractions of a day. * 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. 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

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, pence and farthings, and
ready reckoner A ready reckoner is a printed book or table containing pre-calculated values, often multiples of given amounts. They were widely used in shops and by tradesmen before the advent of cheap electronic calculators, metric weights and measures and ...
s, which are 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 booklet 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 years—in 1586, the Flemish mathematician
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
published a small pamphlet called ''
De Thiende ''De Thiende'', published in 1585 in the Dutch language by Simon Stevin, is remembered for extending positional notation to the use of decimals to represent fractions. A French version, ''La Disme'', was issued the same year by Stevin. Stevin int ...
'' ("the tenth") 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 (e.g., "2.5 ft" is displayed rather than ).


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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
,
divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
, and the solution of equations in integers, such as
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
. 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 In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
,
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ...
,
Diophantine geometry In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to stud ...
and
arithmetic algebraic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...
.
Wiles' proof of Fermat's Last Theorem Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat ...
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 Primary education or elementary education is typically the first stage of formal education, coming after preschool/kindergarten and before secondary school. Primary education takes place in ''primary schools'', ''elementary schools'', or first ...
in mathematics often places a strong focus on algorithms for the arithmetic of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s,
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s,
fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, and
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
s (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 New Mathematics or New Math was a dramatic but temporary change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries and elsewhere, during the 1950s1970s. Curriculum topics and teaching pract ...
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. Also, arithmetic was used by
Islamic Scholars In Islam, the ''ulama'' (; ar, علماء ', singular ', "scholar", literally "the learned ones", also spelled ''ulema''; feminine: ''alimah'' ingularand ''aalimath'' lural are the guardians, transmitters, and interpreters of religious ...
in order to teach application of the rulings related to
Zakat Zakat ( ar, زكاة; , "that which purifies", also Zakat al-mal , "zakat on wealth", or Zakah) is a form of almsgiving, often collected by the Muslim Ummah. It is considered in Islam as a religious obligation, and by Quranic ranking, is n ...
and Irth. This was done in a book entitled ''The Best of Arithmetic'' by Abd-al-Fattah-al-Dumyati. The book begins with the foundations of mathematics and proceeds to its application in the later chapters.


See also

*
Lists of mathematics topics Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. ...
*
Outline of arithmetic Arithmetic is an elementary branch of mathematics that is widely used for tasks ranging from simple day-to-day counting to advanced science and business calculations. Essence of arithmetic *Elementary arithmetic * Decimal arithmetic *Decimal poi ...
*
Slide rule The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which i ...


Related topics

*
Addition of natural numbers Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
*
Additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
*
Arithmetic coding Arithmetic coding (AC) is a form of entropy encoding used in lossless data compression. Normally, a string of characters is represented using a fixed number of bits per character, as in the ASCII code. When a string is converted to arithmetic e ...
*
Arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...
* Arithmetic number *
Arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
*
Arithmetic properties In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying prope ...
*
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
*
Commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name ...
*
Distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
*
Elementary arithmetic The operators in elementary arithmetic are addition, subtraction, multiplication, and division. The operators can be applied on both real numbers and imaginary numbers. Each kind of number is represented on a number line designated to the type. ...
*
Finite field arithmetic In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinite ...
*
Geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
*
Integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
* List of important publications in mathematics * Lunar arithmetic *
Mental calculation Mental calculation consists of arithmetical calculations using only the human brain, with no help from any supplies (such as pencil and paper) or devices such as a calculator. People may use mental calculation when computing tools are not availa ...
*
Number line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
*
Plant arithmetic Plant arithmetic is a form of plant cognition whereby plants appear to perform arithmetic operations – a form of number sense in plants. Arithmetic by species Venus flytrap The Venus flytrap can count to two and five in order to trap and then ...


Notes


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 ''History of the Theory of Numbers'' is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, with little further discussion. T ...
'' (3 volumes), reprints: Carnegie Institute of Washington, Washington, 1932; Chelsea, New York, 1952, 1966 * Euler, Leonhard,
Elements of Algebra
', Tarquin Press, 2007 * Fine, Henry Burchard (1858–1928), ''The Number System of Algebra Treated Theoretically and Historically'', Leach, Shewell & Sanborn, Boston, 1891 * Karpinski, Louis Charles (1878–1956), ''The History of Arithmetic'', Rand McNally, Chicago, 1925; reprint: Russell & Russell, New York, 1965 * Ore, Øystein, ''Number Theory and Its History'', McGraw–Hill, New York, 1948 * Weil, André, ''Number Theory: An Approach through History'', Birkhauser, Boston, 1984; reviewed:
Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...
85c:01004


External links


MathWorld article about arithmetic
* The New Student's Reference Work/Arithmetic (historical)
The Great Calculation According to the Indians, of Maximus Planudes
– an early Western work on arithmetic a
Convergence
* {{Authority control Mathematics education