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Addition (usually signified by the plus symbol ) is one of the four basic operations of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
, 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 ...
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
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'' 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 (that is, "3 ''plus'' 2 is
equal Equal(s) may refer to: Mathematics * Equality (mathematics). * Equals sign (=), a mathematical symbol used to indicate equality. Arts and entertainment * ''Equals'' (film), a 2015 American science fiction film * ''Equals'' (game), a board game ...
to 5"). Besides
counting Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every el ...
items, addition can also be defined and executed without referring to concrete objects, using abstractions called
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s,
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s and
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. Addition belongs to arithmetic, a branch 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 ...
. In
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 a ...
, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and
subgroup In group theory, a branch of mathematics, given a 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, ''H'' is a subgroup ...
s. Addition has several important properties. It 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 ...
, meaning that the order of the
operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above exampl ...
s does not matter, and it is
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 ...
, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see ''
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, mat ...
''). Repeated addition of is the same as counting (see ''
Successor function In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...
''). 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 to do. 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 or elementary education is typically the first stage of formal education, coming after preschool/kindergarten and before secondary school. Primary education takes place in ''primary schools'', ''elementary schools'', or firs ...
, students are taught to add numbers in the
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 ...
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 which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
to the modern
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 pro ...
, where research on the most efficient implementations of addition continues to this day.

Notation and terminology

plus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
"+" between the terms; that is, in
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 a ...
. The result is expressed with an
equals sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between two ...
. 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) :$3 + 3 + 3 + 3 = 12$ (see "multiplication" below) There are also situations where addition is "understood", even though no symbol appears: * A whole number followed immediately by a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
indicates the sum of the two, called a ''mixed number''. For example,$3\frac=3+\frac=3.5.$ 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 mathematical operations of arithmetic, with the other ones being addi ...
instead. The sum of a series 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, :$\sum_^5 k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55.$

Terms

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 multiplied. Some authors call the first addend the ''augend''. and In fact, during the
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 idea ...
, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property 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 branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of ...
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national i ...
words derived from the Latin
verb A verb () is a word (part of speech) that in syntax generally conveys an action (''bring'', ''read'', ''walk'', ''run'', ''learn''), an occurrence (''happen'', ''become''), or a state of being (''be'', ''exist'', ''stand''). In the usual descr ...
''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 stru ...
of ''ad'' "to" and ''dare'' "to give", from the Proto-Indo-European root "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 present active participle. In Late Latin, the differences were larg ...
suffix In linguistics, a suffix is an affix which is placed after the stem of a word. Common examples are case endings, which indicate the grammatical case of nouns, adjectives, and verb endings, which form the conjugation of verbs. Suffixes can carry ...
''-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 that generally functions as the name of a specific object or set of objects, such as living creatures, places, actions, qualities, states of existence, or ideas.Example nouns for: * Living creatures (including people, alive, ...
''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 northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
and Romans 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 Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tra ...
, 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''. He originated the idea that all buildings should have three attribute ...
and
Frontinus Sextus Julius Frontinus (c. 40 – 103 AD) was a prominent 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 Rhine and Danube ...
; Boethius also used several other terms for the addition operation. The later
Middle English Middle English (abbreviated to ME) is a form of the English language that was spoken after the Norman conquest of 1066, until the late 15th century. The English language underwent distinct variations and developments following the Old English p ...
Chaucer Geoffrey Chaucer (; – 25 October 1400) was an English poet, author, and civil servant best known for ''The Canterbury Tales''. He has been called the "father of English literature", or, alternatively, the "father of English poetry". He wa ...
. The
plus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
"+" (
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, w ...
:U+002B;
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
: &#43;) 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
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, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary o ...
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, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
+''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 ...
, because each unary addition operation has an inverse unary subtraction operation, and ''vice versa''.

Properties

Commutativity

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 ...
, 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, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary o ...
s are commutative, such as multiplication, but many others are not, such as subtraction and division.

Associativity

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 ...
, which means that when three or more numbers are added together, the
order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For exampl ...
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 and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For exampl ...
becomes important. In the standard order of operations, addition is a lower priority than
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 r ...
,
nth root In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'': :r^n = x, where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root ...
s, multiplication and division, but is given equal priority to subtraction.

Identity element

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
to any number, does not change the number; this means that zero 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 ...
for addition, and is also known as 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 eleme ...
. In symbols, for every , one has :. This law was first identified in
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
'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 chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...
refined the concept; around the year 830,
Mahavira Mahavira (Sanskrit: महावीर) also known as Vardhaman, was the 24th ''tirthankara'' (supreme preacher) of Jainism. He was the spiritual successor of the 23rd ''tirthankara'' Parshvanatha. Mahavira was born in the early part of the 6 ...
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 also plays a special role: for any integer ''a'', the integer is the least integer greater than ''a'', also known as the successor 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, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
, 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 and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
.

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 decreases after repeated or prolonged presentations of that stimulus. Responses that habituate include those that involve the intact or ...
:
infant An infant or baby is the very young offspring of human beings. ''Infant'' (from the Latin word ''infans'', meaning 'unable to speak' or 'speechless') is a formal or specialised synonym for the common term ''baby''. The terms may also be used ...
s look longer at situations that are unexpected. A seminal experiment by Karen Wynn in 1992 involving
Mickey Mouse Mickey Mouse is an animated cartoon Character (arts), character co-created in 1928 by Walt Disney and Ub Iwerks. The longtime mascot of The Walt Disney Company, Mickey is an Anthropomorphism, anthropomorphic mouse who typically wears red sho ...
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 solid rackets. It takes place on a hard table div ...
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 Primates are a diverse order of mammals. They are divided into the strepsirrhines, which include the lemurs, galagos, and lorisids, and the haplorhines, which include the tarsiers and the simians (monkeys and apes, the latter including ...
s. In a 1995 experiment imitating Wynn's 1992 result (but using
eggplant Eggplant ( US, Canada), aubergine ( UK, Ireland) or brinjal (Indian subcontinent, Singapore, Malaysia, South Africa) is a plant species in the nightshade family Solanaceae. ''Solanum melongena'' is grown worldwide for its edible fruit. Mo ...
rhesus macaque The rhesus macaque (''Macaca mulatta''), colloquially rhesus monkey, is a species of Old World monkey. There are between six and nine recognised subspecies that are split between two groups, the Chinese-derived and the Indian-derived. Generally b ...
and 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 digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
0 through 4, one
chimpanzee The chimpanzee (''Pan troglodytes''), also known as simply the chimp, is a species of great ape native to the forest and savannah of tropical Africa. It has four confirmed subspecies and a fifth proposed subspecies. When its close relative ...
was able to compute the sum of two numerals without further training. More recently,
Asian elephant The Asian elephant (''Elephas maximus''), also known as the Asiatic elephant, is the only living species of the genus ''Elephas'' and is distributed throughout the Indian subcontinent and Southeast Asia, from India in the west, Nepal in the no ...
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 elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every el ...
. 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 bonds"), 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 (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 ...
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 without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...
. * ''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 mathematical operations of arithmetic, with the other ones being addi ...
. 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. Since the end of the 20th century, some US programs, including TERC, decided to remove the traditional transfer method from their curriculum. This decision was criticized, which is why some states and counties didn't support this experiment.

Decimal fractions

Decimal fractions 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 numera ...
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 Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, ...
, 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: :$2.34\times10^ + 5.67\times10^ = 2.34\times10^ + 0.567\times10^ = 2.907\times10^$

Computers

Analog computer An analog computer or analogue computer is a type of Computation, computer that uses the continuous variation aspect of physical phenomena such as Electrical network, electrical, Mechanics, mechanical, or Hydraulics, hydraulic quantities (''a ...
s 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
averaging In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
lever A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is di ...
. If the addends are the rotation speeds of two shafts, they can be added with a differential. A hydraulic adder can add the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and ...
s in two chambers by exploiting Newton's second law to balance forces on an assembly of
piston A piston is a component of reciprocating engines, reciprocating pumps, gas compressors, hydraulic cylinders and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-t ...
s. The most common situation for a general-purpose analog computer is to add two
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge t ...
s (referenced to ground); this can be accomplished roughly with a
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
network Network, networking and networked may refer to: Science and technology * Network theory, the study of graphs as a representation of relations between discrete objects * Network science, an academic field that studies complex networks Mathematic ...
, but a better design exploits an
operational amplifier An operational amplifier (often op amp or opamp) is a DC-coupled high- gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to ...
. Addition is also fundamental to the operation of digital computers, where the efficiency of addition, in particular the 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 which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
, 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 Asia (, ) is one of the world's most notable geographical regions, which is either considered a continent in its own right or a Continent#Subcontinents, subcontinent of Eurasia, which shares the continental landmass of Afro-Eurasia with Afr ...
,
Africa Africa is the world's second-largest and second-most populous continent, after Asia in both cases. At about 30.3 million km2 (11.7 million square miles) including adjacent islands, it covers 6% of Earth's total surface area ...
, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in
Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles o ...
.
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest ...
invented the mechanical calculator in 1642; Jean Marguin, p. 48 (1994) ; Quoting René Taton (1963) it was the first operational
adding machine An adding machine is a class of mechanical calculator, usually specialized for bookkeeping calculations. In the United States, the earliest adding machines were usually built to read in dollars and cents. Adding machines were ubiquitous off ...
. 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 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. Adders execute integer addition in electronic digital computers, usually using
binary arithmetic A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" ( zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the 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 XOR and
AND or AND may refer to: Logic, grammar, and computing * Conjunction (grammar), connecting two words, phrases, or clauses * Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition * Bitwise AND, a boolea ...
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 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 operations as well as such basic tasks as
address An address is a collection of information, presented in a mostly fixed format, used to give the location of a building, apartment, or other structure or a plot of land, generally using political boundaries and street names as references, along w ...
generation during
memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered ...
access and fetching instructions during branching. To increase speed, modern designs calculate digits in parallel; these schemes go by such names as carry select, carry lookahead, and the 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 which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
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 branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of ...
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 A microprocessor is a computer processor where the data processing logic and control is included on a single integrated circuit, or a small number of integrated circuits. The microprocessor contains the arithmetic, logic, and control circ ...
often replaces the augend with the sum but preserves the addend. In a
high-level programming language In computer science, a high-level programming language is a programming language with strong abstraction from the details of the computer. In contrast to low-level programming languages, it may use natural language ''elements'', be easier to use, ...
, 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 or
C++ C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded signific ...
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 Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
occurs, resulting in an incorrect answer. Unanticipated arithmetic overflow is a fairly common cause of 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.

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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s. In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, addition is then extended to progressively larger sets that include the natural numbers: the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s, and the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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
cardinalities In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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 $N\left(A \cup B\right)$. Here, is the union of ''A'' and ''B''. An alternate version of this definition allows ''A'' and ''B'' to possibly overlap and then takes their
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...
, a mechanism that allows common elements to be separated out and therefore counted twice. The other popular definition is recursive: * Let ''n''+ be the 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 theorem on the
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
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 Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
.

Integers

The simplest conception of an integer is that it consists of an
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
(which is a natural number) and a
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or ...
(generally either positive or 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 class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of natural numbers under the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each 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: :$\left(a, b\right)+\left(c, d\right)=\left(a+c,b+d\right).$ 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 any commutative
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
with
cancellation property In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . A ...
. 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 In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of a ...
, with the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
as semigroup operation.

Rational numbers (fractions)

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s 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
denominator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s 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 In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fiel ...
''.

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 In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an ele ...
. 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 Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
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 In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
of rationals, lim ''a''''n''. Addition is defined term by term: * Define $\lim_na_n+\lim_nb_n = \lim_n\left(a_n+b_n\right).$ This definition was first published by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
, 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: :$\left(a+bi\right) + \left(c+di\right) = \left(a+c\right) + \left(b+d\right)i.$ 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 In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
three of whose vertices are ''O'', ''A'' and ''B''. Equivalently, ''X'' is the point such that the
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-colline ...
s with vertices ''O'', ''A'', ''B'', and ''X'', ''B'', ''A'', are 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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
and
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
.

Abstract algebra

Vectors

In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...
is an algebraic structure that allows for adding any two 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: :$\left(a,b\right) + \left(c,d\right) = \left(a+c,b+d\right).$ This addition operation is central to
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mech ...
, in which
velocities Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is ...
,
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by th ...
s and
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...
s are all represented by vectors.

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 In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
, the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus. For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to musical set theory. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in
Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
as the "
exclusive or Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , , ...
" function. A similar "wrap around" operation arises in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, where the sum of two angle measures is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
, which in turn generalizes to addition operations on many-dimensional tori.

General theory

The general theory of abstract algebra allows an "addition" operation to be any
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 ...
and
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 ...
operation on a set. Basic
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
s with such an addition operation include
commutative monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
s and
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s.

Set theory and category theory

A far-reaching generalization of addition of natural numbers is the addition of
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
s and
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
s in set theory. These give two different generalizations of addition of natural numbers to the 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 In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...
operation. In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, disjoint union is seen as a particular case of the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The copr ...
operation, and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
and
wedge sum In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...
, 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 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. ...
.

Arithmetic

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 ...
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 function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X ...
s. 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 Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
can be thought of as 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
, 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 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
: :$e^ = e^a e^b.$ This identity allows multiplication to be carried out by consulting a table of
logarithm 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 o ...
s and computing addition by hand; it also enables multiplication on a
slide rule The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which i ...
. The formula is still a good first-order approximation in the broad context of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
s, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. There are even more generalizations of multiplication than addition. In general, multiplication operations always distribute over addition; this requirement is formalized in the definition of a 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 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 ...
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 An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic di ...
, 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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
. However, it presents a perpetual difficulty in
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
, 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 A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
, 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 In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
, 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: :$a + \max\left(b,c\right) = \max\left(a+b,a+c\right).$ 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 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 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 o ...
: :$\log\left(a+b\right) \approx \max\left(\log a, \log b\right),$ 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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, and taking the " classical limit" as ''h'' tends to zero: :$\max\left(a,b\right) = \lim_h\log\left(e^+e^\right).$ In this sense, the maximum operation is a ''dequantized'' version of addition.

Incrementation, also known as the successor operation, is the addition of to a number.
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, mat ...
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 representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
. An infinite summation is a delicate procedure known as a series.
Counting Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every el ...
a finite set is equivalent to summing 1 over the set. Integration is a kind of "summation" over a
continuum Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal numbe ...
, or more precisely and generally, over a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts ( atlas). One ...
. 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 or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
or superposition of states in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.
Convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
is used to add two independent
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s defined by 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.

In music

Addition is also used in the musical set theory. George Perle gives the following example: “do-mi, re-fa♯, mi♭-sol are different requirements of one interval... or other type of equality... and are connected with the axis of symmetry. Do-mi belongs to the family of symmetrically connected dyads as it is shown further:” ''re'' re♯ mi fa fa♯ sol ''sol♯'' ''re'' do♯ do si la♯ la ''sol♯'' Axis of the pitches are italicized, the axis is defined with the pitch category. Thus, do-mi is a part of an interval family-4 and a part of sum family -2 (at G♯ = 0). A tonal range of Alban Berg’s Lyric Suite is a series of six dyads with their total number being 11. If the line is turned and inverted, then it is with all dyads in total being 6. The total number of successive dyads of a tonal range in Lyric Suite is 11 ''do'' sol re re♯ la♯ ''mi♯'' ''si'' mi la sol♯ do♯ ''fa♯'' Axis of the pitches are italicized, the axis is defined by the dyads (interval 1).

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

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 * * * * * * *