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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 19th c ...
, the other three being subtraction, multiplication and division. 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 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 elem ...
items, addition can also be defined and executed without referring to
concrete object 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. The boundary must be defined and identified by t ...
s, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of 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 subgrou ...
s. Addition has several important properties. It is commutative, meaning that the order of the operands 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''). 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 functi ...
''). 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 first ...
, students are taught to add numbers in the decimal 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 H ...
to the modern computer, where research on the most efficient implementations of addition continues to this day.


Notation and terminology

Addition is written using the plus sign "+" between the terms; that is, in infix notation. The result is expressed with 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 *Ernst von Below (1863–1955), German World War I general *Fred Below ...
) :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 *Ernst von Below (1863–1955), German World War I general *Fred 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 denotes 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 in ...
of related numbers can be expressed through
capital sigma notation 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 ...
, which compactly denotes 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, many authors did not consider the first addend an "addend" at all. Today, due to the
commutative property 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 ...
of addition, "augend" is rarely used, and both terms are generally called addends.Schwartzman p. 19 All of the above terminology derives from Latin. "
Addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
" and "
add Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
" are
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 ...
words derived from the Latin verb ''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 struc ...
of ''ad'' "to" and ''dare'' "to give", from the
Proto-Indo-European root The roots of the reconstructed Proto-Indo-European language (PIE) are basic parts of words that carry a lexical meaning, so-called morphemes. PIE roots usually have verbal meaning like "to eat" or "to run". Roots never occurred alone in the lan ...
"to give"; thus to ''add'' is to ''give to''. Using the gerundive suffix ''-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 ''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 Roman or Romans most often refers to: * Rome, the capital city of Italy * Ancient Rome, Roman civilization from 8th century BC to 5th century AD * Roman people, the people of ancient Rome *''Epistle to the Romans'', shortened to ''Romans'', a let ...
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 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 terms "adden" and "adding" were popularized by
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 w ...
. The plus sign "+" ( Unicode: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 ...
: +) 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 numbers, 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 op ...
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 +''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, because each unary addition operation has an inverse unary subtraction operation, and ''vice versa''.


Properties


Commutativity

Addition is commutative, 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 op ...
s are commutative, such as multiplication, but many others are not, such as subtraction and division.


Associativity

Addition 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 ...
, which means that when three or more numbers are added together, the 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 becomes important. In the standard order of operations, addition is a lower priority than exponentiation, nth roots, multiplication and division, but is given equal priority to subtraction.


Identity element

Adding
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
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 ...
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 elem ...
. 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 tr ...
'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 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 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
also plays a special role: for any integer ''a'', the integer is the least integer greater than ''a'', also known as the
successor Successor may refer to: * An entity that comes after another (see Succession (disambiguation)) Film and TV * ''The Successor'' (film), a 1996 film including Laura Girling * ''The Successor'' (TV program), a 2007 Israeli television program Musi ...
of ''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.


Performing addition


Innate ability

Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation: infants look longer at situations that are unexpected. A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse 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 toddlers, between 18 and 35 months, exploited their development of motor control by allowing them to retrieve 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 primates. 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 ...
s instead of dolls), 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 recogn ...
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 the ...
was able to compute the sum of two numerals without further training. More recently, Asian elephants 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 elem ...
. 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 system is the fluent recall or derivation of the 100 single-digit "addition facts". One could memorize all the facts by
rote Rote can refer to: People * Jason Butler Rote, American TV writer *Kyle Rote (1928–2002), American football player and father of: * Kyle Rote, Jr. (born 1950), American soccer player * Ryan Rote (born 1982), baseball pitcher *Tobin Rote (1928– ...
, 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. 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 numeral ...
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, o ...
, 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 computer An analog computer or analogue computer is a type of computer that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities (''analog signals'') to model the problem being solved. In ...
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 ...
. If the addends are the rotation speeds of two
shafts ''Shafts'' was an English feminist magazine produced by Margaret Sibthorp from 1892 until 1899. Initially published weekly and priced at one penny, its themes included votes for women, women's education, and radical attitudes towards vivisection ...
, they can be added with a differential. A hydraulic adder can add the pressures in two chambers by exploiting
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
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-tig ...
s. The most common situation for a general-purpose analog computer is to add two voltages (referenced to
ground Ground may refer to: Geology * Land, the surface of the Earth not covered by water * Soil, a mixture of clay, sand and organic matter present on the surface of the Earth Electricity * Ground (electricity), the reference point in an electrical c ...
); 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 ...
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 Carry or carrying may refer to: People * Carry (name) Finance * Carried interest (or carry), the share of profits in an investment fund paid to the fund manager * Carry (investment), a financial term: the carry of an asset is the gain or cost of ...
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 H ...
, 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 subcontinent of Eurasia, which shares the continental landmass of Afro-Eurasia with Africa. Asia covers an area ...
,
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.
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 m ...
invented the mechanical calculator in 1642; Jean Marguin, p. 48 (1994) ; Quoting
René Taton René Taton (4 April 1915 – 9 August 2004) was a French author, historian of science, and long co-editor (along with Suzanne Delorme) of the ''Revue d'histoire des sciences''. Life He was born on 4 April 1915 in L'Échelle. He died on 9 ...
(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 ...
generation during memory access and fetching instructions during branching. To increase speed, modern designs calculate digits in
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IB ...
; 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 H ...
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 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 or
C++ C, or c, is the third letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''cee'' (pronounced ), plural ''cees''. History "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 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 19th c ...
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.


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 numbers. In set theory, addition is then extended to progressively larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers. (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(A \cup B). Here, is the
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
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 Successor may refer to: * An entity that comes after another (see Succession (disambiguation)) Film and TV * ''The Successor'' (film), a 1996 film including Laura Girling * ''The Successor'' (TV program), a 2007 Israeli television program Musi ...
of ''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 ...
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), an ...
(which is a natural number) and a sign (generally either
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
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 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
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 more ...
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
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 field ...
''.


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 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(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 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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
is centrally concerned with such generalized operations, and they also appear in set theory 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, cate ...
.


Abstract algebra


Vectors

In linear algebra, a vector space 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: :(a,b) + (c,d) = (a+c,b+d). 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 m ...
, in which velocities,
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 the ...
s and forces 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 bo ...
, 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 Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the ...
. 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 ...
as the " exclusive or" function. A similar "wrap around" operation arises in geometry, 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 con ...
, 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 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 ...
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 a ...
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 ...
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 Transfinite may refer to: * Transfinite number, a number larger than all finite numbers, yet not absolutely infinite * Transfinite induction, an extension of mathematical induction to well-ordered sets ** Transfinite recursion Transfinite inducti ...
. 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, cate ...
, 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 coprodu ...
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 more ...
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 In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
. 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 ...
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 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, the product may still make sense; for example, multiplication by yields the
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
of 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 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 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 ad ...
s, 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 distribute over addition; this requirement is formalized in the definition of a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
. 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 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 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, 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(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
negative infinity In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
. 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 The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
" 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 operation, is the addition of to a number. 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 In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the additive identity. Let a_1, a_2, a_3, ... be a sequence of numbers, and le ...
, 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, usuall ...
. 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 in ...
.
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 elem ...
a finite set is equivalent to summing 1 over the set.
Integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
is a kind of "summation" over a continuum, 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 m ...
. 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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
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 Strategy (from Greek στρατηγία ''stratēgia'', "art of troop leader; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the " ar ...
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.
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 variables 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).


See also

*
Lunar arithmetic Lunar arithmetic, formerly called dismal arithmetic, is a version of arithmetic in which the addition and multiplication operations on digits are defined as the max and min operations. Thus, in lunar arithmetic, :2+7=\max\=7 and 2\times 7 = \min\ ...
* 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 Elementary arithmetic Mathematical notation Articles with example C code