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Addition (usually signified by the
plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an abstract concept arising in mathematics. In the us ...

plus symbol
) is one of the four basic
operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...
of
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, the other three being
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 ...

subtraction
,
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 arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
and
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 ...
. The addition of two whole numbers results in the total amount or ''
sum
sum
'' of those values combined. The example in the adjacent image shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the
mathematical expression In mathematics, an expression or mathematical expression is a finite combination of Glossary of mathematical symbols, symbols that is well-formed formula, well-formed according to rules that depend on the context. Mathematical symbols can desig ...
(that is, "3 ''plus'' 2 is
equal Equal or equals may refer to: Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a board game * The Equals, a British pop group formed in 1965 * "Equal", a 2016 song b ...
to 5"). Besides
counting Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...
items, addition can also be defined and executed without referring to
concrete object A bubble of exhaled gas in water In common usage and classical mechanics, a physical object or physical body (or simply an object or body) is a collection of matter within a defined contiguous boundary in three-dimensional space Three-dimen ...
s, using abstractions called
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
s instead, such as
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s,
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s. Addition belongs to arithmetic, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
. In
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, another area of mathematics, addition can also be performed on abstract objects such as
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
,
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object. Fo ...
, subspaces and
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
s. Addition has several important properties. It is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, meaning that order does not matter, and it is
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see ''
Summation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

Summation
''). Repeated addition of is the same as counting; addition of does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication. Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, , can be performed by infants as young as five months, and even some members of other animal species. In
primary education Primary education is typically the first stage of formal education, coming after preschool/kindergarten and before secondary school. Primary education takes place in primary school, the elementary school or first and middle school depending on ...
, students are taught to add numbers in the
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool that has been in use since ancient times and is still in use today. It was used in the ancient Near East The ancient Near East was the home of e ...

abacus
to the modern
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

computer
, where research on the most efficient implementations of addition continues to this day.


Notation and terminology

Addition is written using the
plus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of sign (mathematics), positive and sign (mathematics), negative, respectively. In addition, represents the operation of addition, which results in a Sum (mat ...
"+" between the terms; that is, in
infix notation Infix notation is the notation commonly used in arithmetical and logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ) ...
. The result is expressed with an
equals sign The equals sign (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar an ...

equals sign
. For example, :1 + 1 = 2 ("one plus one equals two") :2 + 2 = 4 ("two plus two equals four") :1 + 2 = 3 ("one plus two equals three") :5 + 4 + 2 = 11 (see "associativity"
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
) :3 + 3 + 3 + 3 = 12 (see "multiplication"
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
) There are also situations where addition is "understood", even though no symbol appears: * A whole number followed immediately by a
fraction A fraction (from Latin ', "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-fifths ...
indicates the sum of the two, called a ''mixed number''. For example,
3½ = 3 + ½ = 3.5.
This notation can cause confusion, since in most other contexts,
juxtaposition Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc. Speech Juxtaposition in literary terms is the showing ...
denotes
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 arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
instead. The sum of a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
of related numbers can be expressed through capital sigma notation, which compactly denotes
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
. For example, :\sum_^5 k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55. The numbers or the objects to be added in general addition are collectively referred to as the terms, the addends or the summands; this terminology carries over to the summation of multiple terms. This is to be distinguished from ''factors'', which are . Some authors call the first addend the ''augend''. and In fact, during the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in ...

Renaissance
, many authors did not consider the first addend an "addend" at all. Today, due to the
commutative property In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of addition, "augend" is rarely used, and both terms are generally called addends.Schwartzman p. 19 All of the above terminology derives from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

Latin
. "
Addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...
" and " add" are
English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

English
words derived from the Latin
verb A verb () is a word (part of speech) that in syntax conveys an action (''bring'', ''read'', ''walk'', ''run'', ''learn''), an occurrence (''happen'', ''become''), or a state of being (''be'', ''exist'', ''stand''). In the usual description of E ...
''addere'', which is in turn a
compound Compound may refer to: Architecture and built environments * Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall ** Compound (fortification), a version of the above fortified with defensive structu ...
of ''ad'' "to" and ''dare'' "to give", from the
Proto-Indo-European root The roots of the reconstructed Proto-Indo-European language Proto-Indo-European (PIE) is the theorized common ancestor of the Indo-European language family. Its proposed features have been derived by linguistic reconstruction from documented ...
"to give"; thus to ''add'' is to ''give to''. Using the
gerundive In Latin grammar, a gerundive () is a verb form that functions as a verbal adjective. In Classical Latin, the gerundive is distinct in form and function from the gerund and the Latin conjugation#Participles, present active participle. In Late Lati ...
suffix In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languag ...
''-nd'' results in "addend", "thing to be added"."Addend" is not a Latin word; in Latin it must be further conjugated, as in ''numerus addendus'' "the number to be added". Likewise from ''augere'' "to increase", one gets "augend", "thing to be increased". "Sum" and "summand" derive from the Latin
noun A noun () is a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), meaning. In many l ...

noun
''summa'' "the highest, the top" and associated verb ''summare''. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...
and
Romans Roman or Romans usually refers to: *Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, ...
to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends. ''Addere'' and ''summare'' date back at least to
Boethius Anicius Manlius Severinus Boëthius, commonly called Boethius (; also Boetius ; 477 – 524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', and philosopher of the early 6th century. He was born about a ye ...
, if not to earlier Roman writers such as
Vitruvius Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura (''On architecture'', published as ''Ten Books on Architecture'') i ...

Vitruvius
and
Frontinus Sextus Julius Frontinus (c. 40 – 103 AD) was a prominent Roman Empire, Roman civil engineer, author, soldier and senator of the late 1st century AD. He was a successful general under Domitian, commanding forces in Roman Britain, and on the Rhi ...
; Boethius also used several other terms for the addition operation. The later
Middle English Middle English (abbreviated to ME) was a form of the English language English is a West Germanic language of the Indo-European language family The Indo-European languages are a language family A language is a structured sys ...
terms "adden" and "adding" were popularized by
Chaucer Geoffrey Chaucer (; – 25 October 1400) was an English poet and author. Widely considered the greatest English poet of the Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th ...

Chaucer
. The
plus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of sign (mathematics), positive and sign (mathematics), negative, respectively. In addition, represents the operation of addition, which results in a Sum (mat ...

plus sign
"+" (
Unicode Unicode, formally the Unicode Standard, is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's wri ...

Unicode
:U+002B;
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding Character encoding is the process of assigning numbers to graphical Graphics (from Greek Greek may refer to: Greece Anything of, ...
: +) is an abbreviation of the Latin word ''et'', meaning "and". It appears in mathematical works dating back to at least 1489.


Interpretations

Addition is used to model many physical processes. Even for the simple case of adding
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s, there are many possible interpretations and even more visual representations.


Combining sets

Possibly the most fundamental interpretation of addition lies in combining sets: * When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections. This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires, see below). However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.


Extending a length

A second interpretation of addition comes from extending an initial length by a given length: * When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension. The sum ''a'' + ''b'' can be interpreted as a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
that combines ''a'' and ''b'', in an algebraic sense, or it can be interpreted as the addition of ''b'' more units to ''a''. Under the latter interpretation, the parts of a sum play asymmetric roles, and the operation is viewed as applying the
unary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
+''b'' to ''a''. Instead of calling both ''a'' and ''b'' addends, it is more appropriate to call ''a'' the augend in this case, since ''a'' plays a passive role. The unary view is also useful when discussing
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 ...

subtraction
, because each unary addition operation has an inverse unary subtraction operation, and ''vice versa''.


Properties


Commutativity

Addition is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if ''a'' and ''b'' are any two numbers, then :''a'' + ''b'' = ''b'' + ''a''. The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are commutative, such as multiplication, but many others are not, such as subtraction and division.


Associativity

Addition is
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, which means that when three or more numbers are added together, the
order of operations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

order of operations
does not change the result. As an example, should the expression ''a'' + ''b'' + ''c'' be defined to mean (''a'' + ''b'') + ''c'' or ''a'' + (''b'' + ''c'')? Given that addition is associative, the choice of definition is irrelevant. For any three numbers ''a'', ''b'', and ''c'', it is true that . For example, . When addition is used together with other operations, the
order of operations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

order of operations
becomes important. In the standard order of operations, addition is a lower priority than
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
,
nth root In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, multiplication and division, but is given equal priority to subtraction.


Identity element

When adding
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
to any number, the quantity does not change; zero is the
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
for addition, also known as the
additive identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. In symbols, for any ''a'', :''a'' + 0 = 0 + ''a'' = ''a''. This law was first identified in
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

Brahmagupta
's '' Brahmasphutasiddhanta'' in 628 AD, although he wrote it as three separate laws, depending on whether ''a'' is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later
Indian mathematicians The chronology of Indian mathematicians spans from the Indus Valley Civilization and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematici ...
refined the concept; around the year 830,
Mahavira Mahavira ( sa, महावीर:), also known as Vardhamana, was the 24th ''Tirthankara In Jainism Jainism (), traditionally known as ''Jain Dharma'', is an ancient Indian religion and the method of acquiring perfect knowledge ...
wrote, "zero becomes the same as what is added to it", corresponding to the unary statement . In the 12th century, Bhaskara wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement .


Successor

Within the context of integers, addition of
one 1 (one, also called unit, and unity) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can ...

one
also plays a special role: for any integer ''a'', the integer is the least integer greater than ''a'', also known as the
successor Successor is someone who, or something which succeeds or comes after (see success (disambiguation), success and Succession (disambiguation), succession) Film and TV * The Successor (film), ''The Successor'' (film), a 1996 film including Laura Girli ...
of ''a''. For instance, 3 is the successor of 2 and 7 is the successor of 6. Because of this succession, the value of can also be seen as the ''b''th successor of ''a'', making addition iterated succession. For example, is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the 2nd successor of 6.


Units

To numerically add physical quantities with
units Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
, they must be expressed with common units. For example, adding 50 milliliters to 150 milliliters gives 200 milliliters. However, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in
dimensional analysis In engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range ...
.


Performing addition


Innate ability

Studies on mathematical development starting around the 1980s have exploited the phenomenon of
habituation Habituation is a form of non-associative learning in which an innate (non-reinforced) response to a stimulus A stimulus is something that causes a physiological response. It may refer to: *Stimulation Stimulation is the encouragement of deve ...

habituation
:
infant An infant (from the Latin word ''infans'', meaning 'unable to speak' or 'speechless') is the more formal or specialised synonym for the common term ''baby'', meaning the very young offspring of human beings Humans (''Homo sapiens'' ...

infant
s look longer at situations that are unexpected. A seminal experiment by Karen Wynn in 1992 involving
Mickey Mouse Mickey Mouse is a cartoon A cartoon is a type of illustration that is typically drawn, sometimes animated, in an unrealistic or semi-realistic style. The specific meaning has evolved over time, but the modern usage usually refers to ei ...
dolls manipulated behind a screen demonstrated that five-month-old infants ''expect'' to be 2, and they are comparatively surprised when a physical situation seems to imply that is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies. Another 1992 experiment with older
toddler A toddler is a child approximately 12 to 36 months old, though definitions vary. The toddler years are a time of great cognitive, emotional and social development. The word is derived from "to toddle", which means to walk unsteadily, like a child ...
s, between 18 and 35 months, exploited their development of motor control by allowing them to retrieve
ping-pong Table tennis, also known as ping-pong and whiff-whaff, is a sport in which two or four players hit a lightweight ball, also known as the ping-pong ball, back and forth across a table using small rackets. The game takes place on a hard table divi ...

ping-pong
balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5. Even some nonhuman animals show a limited ability to add, particularly
primate A primate ( ) (from Latin , from 'prime, first rank') is a eutherian mammal constituting the Taxonomy (biology), taxonomic order (biology), order Primates (). Primates arose 85–55 million years ago first from small Terrestrial animal, ...

primate
s. In a 1995 experiment imitating Wynn's 1992 result (but using
eggplant Eggplant ( US, Australia Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasmania, and numerous ...

eggplant
s instead of dolls),
rhesus macaque The rhesus macaque (''Macaca mulatta''), colloquially rhesus monkey, is a species of Old World monkey Old World monkey is the common English name for a family In , family (from la, familia) is a of people related either by (by recog ...

rhesus macaque
and
cottontop tamarin The cotton-top tamarin (''Saguinus oedipus'') is a small New World monkey weighing less than . This New World monkey can live up to 24 years, but most of them die by 13 years. One of the smallest primates, the cotton-top tamarin is easily recogni ...

cottontop tamarin
monkeys performed similarly to human infants. More dramatically, after being taught the meanings of the
Arabic numerals Arabic numerals are the ten numerical digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a Positional notation, positional numeral sy ...

Arabic numerals
0 through 4, one
chimpanzee The chimpanzee (''Pan troglodytes''), also known simply as chimp, is a species of Hominidae, great ape native to the forest and savannah of tropical Africa. It has four confirmed subspecies and a fifth proposed subspecies. The chimpanzee and t ...
was able to compute the sum of two numerals without further training. More recently, s have demonstrated an ability to perform basic arithmetic.


Childhood learning

Typically, children first master
counting Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...
. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, ''five''" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers. Most discover it independently. With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting "four, ''five''." Eventually children begin to recall certain addition facts ("
number bondIn mathematics education at primary school level, a number bond (sometimes alternatively called an addition fact) is a simple addition sum which has become so familiar that a child can recognise it and complete it almost instantly, with recall as aut ...
s"), either through experience or rote memorization. Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that and then reason that is one more, or 13. Such derived facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and derived facts to add fluently. Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school. However, throughout the world, addition is taught by the end of the first year of elementary school.


Table

Children are often presented with the addition table of pairs of numbers from 0 to 9 to memorize. Knowing this, children can perform any addition.


Decimal system

The prerequisite to addition in the
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
system is the fluent recall or derivation of the 100 single-digit "addition facts". One could memorize all the facts by rote, but pattern-based strategies are more enlightening and, for most people, more efficient:Fosnot and Dolk p. 99 * ''Commutative property'': Mentioned above, using the pattern ''a + b = b + a'' reduces the number of "addition facts" from 100 to 55. * ''One or two more'': Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately,
intuition Intuition is the ability to acquire knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is ...
. * ''Zero'': Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends; word problems may help rationalize the "exception" of zero. * ''Doubles'': Adding a number to itself is related to counting by two and to
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 arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp. * ''Near-doubles'': Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact by adding one more, or from but subtracting one. * ''Five and ten'': Sums of the form 5 + and 10 + are usually memorized early and can be used for deriving other facts. For example, can be derived from by adding one more. * ''Making ten'': An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, . As students grow older, they commit more facts to memory, and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still find any basic fact quickly.


Carry

The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from the ones column on the right. If a column exceeds nine, the extra digit is " carried" into the next column. For example, in the addition ¹ 27 + 59 ———— 86 7 + 9 = 16, and the digit 1 is the carry.Some authors think that "carry" may be inappropriate for education; Van de Walle (p. 211) calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term. An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum. There are many alternative methods.


Decimal fractions

Decimal fractions The decimal numeral system (also called the base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the H ...
can be added by a simple modification of the above process. One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands. As an example, 45.1 + 4.34 can be solved as follows: 4 5 . 1 0 + 0 4 . 3 4 ———————————— 4 9 . 4 4


Scientific notation

In scientific notation, numbers are written in the form x=a\times10^, where a is the significand and 10^ is the exponential part. Addition requires two numbers in scientific notation to be represented using the same exponential part, so that the two significands can simply be added. For example: :2.34\times10^ + 5.67\times10^ = 2.34\times10^ + 0.567\times10^ = 2.907\times10^


Non-decimal

Addition in other bases is very similar to decimal addition. As an example, one can consider addition in binary. Adding two single-digit binary numbers is relatively simple, using a form of carrying: :0 + 0 → 0 :0 + 1 → 1 :1 + 0 → 1 :1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21)) Adding two "1" digits produces a digit "0", while 1 must be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented: :5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101)) :7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101)) This is known as ''carrying''. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary: 0 1 1 0 1 + 1 0 1 1 1 ————————————— 1 0 0 1 0 0 = 36 In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, . The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: again; the 1 is carried, and 0 is written at the bottom. The third column: . This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).


Computers

Analog computers work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with an arithmetic mean, averaging lever. If the addends are the rotation speeds of two axle, shafts, they can be added with a differential (mechanics), differential. A hydraulic adder can add the pressures in two chambers by exploiting Newton's laws of motion, Newton's second law to balance forces on an assembly of pistons. The most common situation for a general-purpose analog computer is to add two voltages (referenced to ground (electricity), ground); this can be accomplished roughly with a resistor Electronic circuit, network, but a better design exploits an operational amplifier. Addition is also fundamental to the operation of computer, digital computers, where the efficiency of addition, in particular the carry (arithmetic), carry mechanism, is an important limitation to overall performance. The
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool that has been in use since ancient times and is still in use today. It was used in the ancient Near East The ancient Near East was the home of e ...

abacus
, also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia, Africa, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in Sumer. Blaise Pascal invented the mechanical calculator in 1642;Addition#MARG, Jean Marguin, p. 48 (1994) ; Quoting Addition#TATON63, René Taton (1963) it was the first operational adding machine. It made use of a gravity-assisted carry mechanism. It was the only operational mechanical calculator in the 17th century and the earliest automatic, digital computer. Pascal's calculator was limited by its carry mechanism, which forced its wheels to only turn one way so it could add. To subtract, the operator had to use the method of complements, Pascal's calculator's complement, which required as many steps as an addition. Giovanni Poleni followed Pascal, building the second functional mechanical calculator in 1709, a calculating clock made of wood that, once setup, could multiply two numbers automatically. Adder (electronics), Adders execute integer addition in electronic digital computers, usually using binary arithmetic. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the Carry bypass adder, carry skip design, again following human intuition; one does not perform all the carries in computing , but one bypasses the group of 9s and skips to the answer. In practice, computational addition may be achieved via Exclusive or, XOR and Bitwise operation#AND, AND bitwise logical operations in conjunction with bitshift operations as shown in the pseudocode below. Both XOR and AND gates are straightforward to realize in digital logic allowing the realization of Adder (electronics), full adder circuits which in turn may be combined into more complex logical operations. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all floating point, floating-point operations as well as such basic tasks as memory address, address generation during memory (computers), memory access and fetching instruction (computer science), instructions during control flow, branching. To increase speed, modern designs calculate digits in parallel algorithm, parallel; these schemes go by such names as carry select, carry lookahead adder, carry lookahead, and the Ling adder, Ling pseudocarry. Many implementations are, in fact, hybrids of these last three designs. Unlike addition on paper, addition on a computer often changes the addends. On the ancient
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool that has been in use since ancient times and is still in use today. It was used in the ancient Near East The ancient Near East was the home of e ...

abacus
and adding board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

Latin
texts often claimed that in the process of adding "a number to a number", both numbers vanish. In modern times, the ADD instruction of a microprocessor often replaces the augend with the sum but preserves the addend. In a high-level programming language, evaluating does not change either ''a'' or ''b''; if the goal is to replace ''a'' with the sum this must be explicitly requested, typically with the statement . Some languages such as C (programming language), C or C++ allow this to be abbreviated as . // Iterative algorithm int add(int x, int y) // Recursive algorithm int add(int x, int y) On a computer, if the result of an addition is too large to store, an arithmetic overflow occurs, resulting in an incorrect answer. Unanticipated arithmetic overflow is a fairly common cause of software bug, program errors. Such overflow bugs may be hard to discover and diagnose because they may manifest themselves only for very large input data sets, which are less likely to be used in validation tests. The Year 2000 problem was a series of bugs where overflow errors occurred due to use of a 2-digit format for years.


Addition of numbers

To prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined on the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s. In set theory, addition is then extended to progressively larger sets that include the natural numbers: the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, the rational numbers, and the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. (In mathematics education, positive fractions are added before negative numbers are even considered; this is also the historical route.)


Natural numbers

There are two popular ways to define the sum of two natural numbers ''a'' and ''b''. If one defines natural numbers to be the Cardinal number, cardinalities of finite sets, (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows: * Let N(''S'') be the cardinality of a set ''S''. Take two disjoint sets ''A'' and ''B'', with and . Then is defined as N(A \cup B). Here, is the union (set theory), union of ''A'' and ''B''. An alternate version of this definition allows ''A'' and ''B'' to possibly overlap and then takes their disjoint union, a mechanism that allows common elements to be separated out and therefore counted twice. The other popular definition is recursive: * Let ''n''+ be the Peano axioms#Binary operations and ordering, successor of ''n'', that is the number following ''n'' in the natural numbers, so 0+=1, 1+=2. Define . Define the general sum recursively by . Hence . Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the Recursion#The recursion theorem, recursion theorem on the partially ordered set N2. On the other hand, some sources prefer to use a restricted recursion theorem that applies only to the set of natural numbers. One then considers ''a'' to be temporarily "fixed", applies recursion on ''b'' to define a function "''a'' +", and pastes these unary operations for all ''a'' together to form the full binary operation. This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, through mathematical induction.


Integers

The simplest conception of an integer is that it consists of an absolute value (which is a natural number) and a sign (mathematics), sign (generally either positive number, positive or negative numbers, negative). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases: * For an integer ''n'', let , ''n'', be its absolute value. Let ''a'' and ''b'' be integers. If either ''a'' or ''b'' is zero, treat it as an identity. If ''a'' and ''b'' are both positive, define . If ''a'' and ''b'' are both negative, define . If ''a'' and ''b'' have different signs, define to be the difference between , ''a'', and , ''b'', , with the sign of the term whose absolute value is larger. As an example, ; because −6 and 4 have different signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative. Although this definition can be useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. So the following method is commonly used for defining integers. It is based on the remark that every integer is the difference of two natural integers and that two such differences, and are equal if and only if . So, one can define formally the integers as the equivalence classes of ordered pairs of natural numbers under the equivalence relation : if and only if . The equivalence class of contains either if , or otherwise. If is a natural number, one can denote the equivalence class of , and by the equivalence class of . This allows identifying the natural number with the equivalence class . Addition of ordered pairs is done component-wise: : (a, b)+(c, d)=(a+c,b+d). A straightforward computation shows that the equivalence class of the result depends only on the equivalences classes of the summands, and thus that this defines an addition of equivalence classes, that is integers. Another straightforward computation shows that this addition is the same as the above case definition. This way of defining integers as equivalence classes of pairs of natural numbers, can be used to embed into a group (mathematics), group any commutative semigroup with cancellation property. Here, the semigroup is formed by the natural numbers and the group is the additive group of integers. The rational numbers are constructed similarly, by taking as semigroup the nonzero integers with multiplication. This construction has been also generalized under the name of Grothendieck group to the case of any commutative semigroup. Without the cancellation property the semigroup homomorphism from the semigroup into the group may be non-injective. Originally, the ''Grothendieck group'' was, more specifically, the result of this construction applied to the equivalences classes under isomorphisms of the objects of an abelian category, with the direct sum as semigroup operation.


Rational numbers (fractions)

Addition of rational numbers can be computed using the least common denominator, but a conceptually simpler definition involves only integer addition and multiplication: * Define \frac ab + \frac cd = \frac. As an example, the sum \frac 34 + \frac 18 = \frac = \frac = \frac = \frac78. Addition of fractions is much simpler when the denominators are the same; in this case, one can simply add the numerators while leaving the denominator the same: \frac ac + \frac bc = \frac, so \frac 14 + \frac 24 = \frac = \frac 34. The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic. For a more rigorous and general discussion, see ''field of fractions''.


Real numbers

A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a Dedekind cut of rationals: a non-empty set of rationals that is closed downward and has no greatest element. The sum of real numbers ''a'' and ''b'' is defined element by element: * Define a+b = \. This definition was first published, in a slightly modified form, by Richard Dedekind in 1872. The commutativity and associativity of real addition are immediate; defining the real number 0 to be the set of negative rationals, it is easily seen to be the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses. Unfortunately, dealing with multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers. Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of a Cauchy sequence of rationals, lim ''a''''n''. Addition is defined term by term: * Define \lim_na_n+\lim_nb_n = \lim_n(a_n+b_n). This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different. One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions.


Complex numbers

Complex numbers are added by adding the real and imaginary parts of the summands. That is to say: :(a+bi) + (c+di) = (a+c) + (b+d)i. Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers ''A'' and ''B'', interpreted as points of the complex plane, is the point ''X'' obtained by building a parallelogram three of whose vertices are ''O'', ''A'' and ''B''. Equivalently, ''X'' is the point such that the triangles with vertices ''O'', ''A'', ''B'', and ''X'', ''B'', ''A'', are Congruence (geometry), congruent.


Generalizations

There are many binary operations that can be viewed as generalizations of the addition operation on the real numbers. The field of abstract algebra is centrally concerned with such generalized operations, and they also appear in set theory and category theory.


Abstract algebra


Vectors

In linear algebra, a vector space is an algebraic structure that allows for adding any two coordinate vector, vectors and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (''a'',''b'') is interpreted as a vector from the origin in the Euclidean plane to the point (''a'',''b'') in the plane. The sum of two vectors is obtained by adding their individual coordinates: :(a,b) + (c,d) = (a+c,b+d). This addition operation is central to classical mechanics, in which vectors are interpreted as forces.


Matrices

Matrix addition is defined for two matrices of the same dimensions. The sum of two ''m'' × ''n'' (pronounced "m by n") matrices A and B, denoted by , is again an matrix computed by adding corresponding elements: :\begin \mathbf+\mathbf & = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \\ \end + \begin b_ & b_ & \cdots & b_ \\ b_ & b_ & \cdots & b_ \\ \vdots & \vdots & \ddots & \vdots \\ b_ & b_ & \cdots & b_ \\ \end \\ & = \begin a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ \end \\ \end For example: : \begin 1 & 3 \\ 1 & 0 \\ 1 & 2 \end + \begin 0 & 0 \\ 7 & 5 \\ 2 & 1 \end = \begin 1+0 & 3+0 \\ 1+7 & 0+5 \\ 1+2 & 2+1 \end = \begin 1 & 3 \\ 8 & 5 \\ 3 & 3 \end


Modular arithmetic

In modular arithmetic, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to set theory (music), musical set theory. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the "exclusive or" function. In geometry, the sum of two angle, angle measures is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the circle, which in turn generalizes to addition operations on many-dimensional torus, tori.


General theory

The general theory of abstract algebra allows an "addition" operation to be any associative and
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
operation on a set. Basic algebraic structures with such an addition operation include commutative monoids and abelian groups.


Set theory and category theory

A far-reaching generalization of addition of natural numbers is the addition of ordinal numbers and cardinal numbers in set theory. These give two different generalizations of addition of natural numbers to the transfinite number, transfinite. Unlike most addition operations, addition of ordinal numbers is not commutative. Addition of cardinal numbers, however, is a commutative operation closely related to the disjoint union operation. In category theory, disjoint union is seen as a particular case of the coproduct operation, and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as direct sum and wedge sum, are named to evoke their connection with addition.


Related operations

Addition, along with subtraction, multiplication and division, is considered one of the basic operations and is used in elementary arithmetic.


Arithmetic

Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding and subtracting are inverse functions. Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that is closed under subtraction. Multiplication can be thought of as Multiplication and repeated addition, repeated addition. If a single term appears in a sum ''n'' times, then the sum is the product of ''n'' and . If ''n'' is not a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
, the product may still make sense; for example, multiplication by yields the additive inverse of a number. In the real and complex numbers, addition and multiplication can be interchanged by the exponential function: :e^ = e^a e^b. This identity allows multiplication to be carried out by consulting a mathematical table, table of logarithms and computing addition by hand; it also enables multiplication on a slide rule. The formula is still a good first-order approximation in the broad context of Lie groups, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated Lie algebra. There are even more generalizations of multiplication than addition. In general, multiplication operations always distributivity, distribute over addition; this requirement is formalized in the definition of a ring (mathematics), ring. In some contexts, such as the integers, distributivity over addition and the existence of a multiplicative identity is enough to uniquely determine the multiplication operation. The distributive property also provides information about addition; by expanding the product in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general. Division (mathematics), Division is an arithmetic operation remotely related to addition. Since , division is right distributive over addition: . However, division is not left distributive over addition; is not the same as .


Ordering

The maximum operation "max (''a'', ''b'')" is a binary operation similar to addition. In fact, if two nonnegative numbers ''a'' and ''b'' are of different orders of magnitude, then their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example in truncating Taylor series. However, it presents a perpetual difficulty in numerical analysis, essentially since "max" is not invertible. If ''b'' is much greater than ''a'', then a straightforward calculation of can accumulate an unacceptable round-off error, perhaps even returning zero. See also ''Loss of significance''. The approximation becomes exact in a kind of infinite limit; if either ''a'' or ''b'' is an infinite cardinal number, their cardinal sum is exactly equal to the greater of the two. Accordingly, there is no subtraction operation for infinite cardinals. Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition: :a + \max(b,c) = \max(a+b,a+c). For these reasons, in tropical geometry one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is extended real number line, negative infinity. Some authors prefer to replace addition with minimization; then the additive identity is positive infinity. Tying these observations together, tropical addition is approximately related to regular addition through the logarithm: :\log(a+b) \approx \max(\log a, \log b), which becomes more accurate as the base of the logarithm increases. The approximation can be made exact by extracting a constant ''h'', named by analogy with Planck's constant from quantum mechanics, and taking the "classical limit" as ''h'' tends to zero: :\max(a,b) = \lim_h\log(e^+e^). In this sense, the maximum operation is a ''dequantized'' version of addition.


Other ways to add

Incrementation, also known as the Successor function, successor operation, is the addition of to a number.
Summation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

Summation
describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of the sum of a single number, which is itself, and the empty sum, which is
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
. An infinite summation is a delicate procedure known as a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
.Stewart p. 8 Counting a finite set is equivalent to summing 1 over the set. Integral, Integration is a kind of "summation" over a Continuum (set theory), continuum, or more precisely and generally, over a differentiable manifold. Integration over a zero-dimensional manifold reduces to summation. Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier, usually a real numbers, real or complex numbers, complex number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixed strategy, mixing of strategy (game theory), strategies in game theory or quantum superposition, superposition of quantum state, states in quantum mechanics. Convolution is used to add two independent random variables defined by probability distribution, distribution functions. Its usual definition combines integration, subtraction, and multiplication. In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.


See also

* Mental calculation, Mental arithmetic * Parallel addition (mathematics) * Verbal arithmetic (also known as cryptarithms), puzzles involving addition


Notes


Footnotes


References

History * * * * Elementary mathematics * Education *
California State Board of Education mathematics content standards
Adopted December 1997, accessed December 2005. * * * Cognitive science * * Mathematical exposition * * * * * * Advanced mathematics * * * * * * * * Mathematical research * * * Litvinov, Grigory; Maslov, Victor; Sobolevskii, Andreii (1999)
Idempotent mathematics and interval analysis

Reliable Computing
', Kluwer. * * * Computing * * * * * *


Further reading

* * * * * {{Authority control Addition, Elementary arithmetic Mathematical notation