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Arithmetic () is an elementary part of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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'' ...
,
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 Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
,
division,
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
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
. In the 19th century, Italian mathematician
Giuseppe Peano 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 forma ...
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 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 Co ...
, 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 c. 1 ...
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 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 numeral ...
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 th ...
. Complex calculations with Roman numerals required the assistance of a
counting board (or the
Roman abacus) 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, and the
vigesimal
vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'.
Places
In a ...
(base 20) system that defined
Maya numerals. 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 ...
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 bo ...
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 scientis ...
,
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 th ...
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 Multiplication, multiplying digits to the left of 0 by th ...
, 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 ...
dated back to 2nd century BC.
The gradual development of the
Hindu–Arabic numeral system 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 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 tr ...
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. His contemporary, the
Syriac 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 Sy ...
(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''.
![Leibniz Stepped Reckoner](https://upload.wikimedia.org/wikipedia/commons/b/b8/Leibniz_Stepped_Reckoner.png)
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 Wester ...
) was primarily responsible for spreading their use throughout Europe after the publication of his book ''
Liber Abaci'' 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 Multiplication, multiplying digits to the left of 0 by th ...
".
In the Middle Ages, arithmetic was one of the seven
liberal arts
Liberal arts education (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as La ...
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 a ...
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
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...
, 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 numeral ...
notation.
Various types of tools have been invented and widely used to assist in numeric calculations. Before Renaissance, they were various types of
abaci. More recent examples include
slide rules,
nomograms 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 w ...
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 use ...
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 .
...
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 function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s, 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''. Particul ...
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) can be performed, and where these four operations obey the usual laws (including distributivity), is called a
field.
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, ma ...
, a term also used to denote the definition for "adding infinitely many numbers" in an
infinite series. Repeated addition of the number
1 is the most basic form of
counting; the result of adding is usually called the
successor 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 replacement ...
, 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 su ...
of addition, or the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elemen ...
.
For every number , there is a number denoted , called the ''
opposite'' of , such that and . So, the opposite of is the
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
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 (op ...
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 o ...
nor
associative. 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 on ...
, delivering the additive inverse for any given number, and losing the immediate access to the notion of
difference
Difference, The Difference, Differences or Differently may refer to:
Music
* ''Difference'' (album), by Dreamtale, 2005
* ''Differently'' (album), by Cassie Davis, 2009
** "Differently" (song), by Cassie Davis, 2009
* ''The Difference'' (al ...
, 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
or
, 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 is 1, since multiplying any number by 1 yields that same number. The multiplicative inverse for any number except is the reciprocal (mathematics), 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 (mathematics), group 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 Mathematical table, tables or slide rules, 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
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 division by zero, 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 multiplicative inverse, reciprocal of the divisor as factors, that is,
Within the natural numbers, there is also a different but related notion called Euclidean division, 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, denoted by the symbol
or the word
, though sometimes a second output for one "divmod" operation. In either case, Modular arithmetic 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, Euclid's ''Elements'' first introduced this theorem, and gave a partial proof (which is called Euclid's lemma). The fundamental theorem of arithmetic was first proven by Carl Friedrich Gauss.
The fundamental theorem of arithmetic is one of the reasons Prime number#Primality of one, why 1 is not considered a prime number. Other reasons include the sieve of Eratosthenes, 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 as the
digits for a radix decimal, 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 th ...
; 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 Numerals, Greek, Cyrillic numerals, Cyrillic, Roman numerals, Roman, or Chinese numerals 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 tr ...
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 numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent arithmetic fraction, fractions (e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (10
2), plus 0 tens (10
1), plus 7 units (10
0), plus 3 tenths (10
−1) plus 6 hundredths (10
−2).
The concept of Zero, 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 Jainism, Jain text from India entitled the ''Lokavibhâga'', dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the Mathematics in medieval Islam, scholarship of the Arabic world, were introduced into
Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...
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 Wester ...
using the Hindu–Arabic numeral system.
Algorism 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 exponentiation, exponent for the radix (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 ''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 (mathematics), 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.
Compound unit arithmetic
Compound unit arithmetic is the application of arithmetic operations to mixed radix 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 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 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.
![MixedUnitAddition](https://upload.wikimedia.org/wikipedia/commons/9/92/MixedUnitAddition.svg)
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
![Yarloop wkshop gnangarra 14](https://upload.wikimedia.org/wikipedia/commons/b/b0/Yarloop_wkshop_gnangarra_14.jpg)
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 cash register, mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pence and farthings, and ready reckoners, 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 published a small pamphlet called ''De Thiende'' ("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 prime number, primality, divisibility, and the Diophantine equation, solution of equations in integers, such as Fermat's Last Theorem. It appeared that most of these problems, although very elementary to state, are very difficult and may not be solved without very deep mathematics involving concepts and methods from many other branches of mathematics. This led to new branches of number theory such as analytic number theory, algebraic number theory, Diophantine geometry and arithmetic algebraic geometry. Wiles' proof of Fermat's Last Theorem is a typical example of the necessity of sophisticated methods, which go far beyond the classical methods of arithmetic, for solving problems that can be stated in elementary arithmetic.
Arithmetic in education
Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, arithmetic fraction, fractions, 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 numeral ...
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 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 Ulama, Islamic Scholars in order to teach application of the rulings related to Zakat and Islamic inheritance jurisprudence, 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
* Outline of arithmetic
* Slide rule
Related topics
* Addition of natural numbers
* Additive inverse
* Arithmetic coding
* Arithmetic mean
* Arithmetic number
* Arithmetic progression
* Arithmetic properties
* Associativity
* Commutativity
* Distributivity
* Elementary arithmetic
* Finite field arithmetic
* Geometric progression
* Integer
* List of important publications in mathematics
* Lunar arithmetic
* Mental calculation
* Number line
* Plant arithmetic
Notes
References
* Cunnington, Susan, ''The Story of Arithmetic: A Short History of Its Origin and Development'', Swan Sonnenschein, London, 1904
* Leonard Eugene Dickson, Dickson, Leonard Eugene, ''History of the Theory of Numbers'' (3 volumes), reprints: Carnegie Institute of Washington, Washington, 1932; Chelsea, New York, 1952, 1966
* Leonhard Euler, Euler, Leonhard,
Elements of Algebra', Tarquin Press, 2007
* Henry Burchard Fine, Fine, Henry Burchard (1858–1928), ''The Number System of Algebra Treated Theoretically and Historically'', Leach, Shewell & Sanborn, Boston, 1891
* Louis Charles Karpinski, Karpinski, Louis Charles (1878–1956), ''The History of Arithmetic'', Rand McNally, Chicago, 1925; reprint: Russell & Russell, New York, 1965
* Øystein Ore, Ore, Øystein, ''Number Theory and Its History'', McGraw–Hill, New York, 1948
* André Weil, Weil, André, ''Number Theory: An Approach through History'', Birkhauser, Boston, 1984; reviewed: Mathematical Reviews 85c:01004
External links
MathWorld article about arithmetic* :s:The New Student's Reference Work/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
Arithmetic,
Mathematics education