Addition is a mathematical operation In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values, that represents combining collections of objects together into a larger collection. It is signified by the plus sign The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning "more" and "less", respectively (+). For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples. Therefore, 3 + 2 = 5. Besides counts of fruit, addition can also represent combining other physical and abstract quantities using different kinds of numbers: negative numbers Being negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as −, −1.414, −1. A positive number is one that is greater than zero, such as , 1.414, 1. Zero itself is neither positive nor negative. The, fractions A fraction is a number that can represent part of a whole, irrational numbers In mathematics, an irrational number is any real number which cannot be expressed as a fraction p/q, where p and q are integers, with q non-zero and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction. It can be proven that irrational numbers are precisely those real, vectors In elementary mathematics, physics, and engineering, a Euclidean vector is a geometric object that has both a magnitude (or length) and direction. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by, decimals and more.

Addition follows several important patterns. It is commutative In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the, meaning that order does not matter, and it is associative In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such, meaning that when one adds more than two numbers, order in which addition is performed does not matter (see Summation Summation is the operation of combining a sequence of numbers using addition; the result is their sum or total. An interim or present total of a summation process is termed the running total. The numbers to be summed may be integers, rational numbers, real numbers, or complex numbers, and other types of values than numbers can be added as well:). Repeated addition of 1 1 is a number, numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement. For example, a line segment of "unit length" is a line segment of length 1 is the same as counting Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left;; addition of 0 Zero, written 0, is both a number and the numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. In the English language, 0 may be called zero, oh, does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with. Subtraction is denoted by a minus sign in infix notation and multiplication Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division). All of these rules can be proven In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to, starting with the addition of natural numbers Addition of natural numbers is the most basic arithmetic binary operation. The operation addition takes two natural numbers, the augend and addend, and produces a single number, the sum. The set of natural numbers will be denoted by N, and "0" will be used to denote the natural number which is not the successor of any other natural and generalizing up through the real numbers In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an and beyond. General binary operations In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division that continue these patterns are studied in abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae.

Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some animals. In primary education Primary education is the first stage of compulsory education. It is preceded by pre-school or nursery education and is followed by secondary education. In North America this stage of education is usually known as elementary education and is generally followed by middle school, children learn to add numbers in the decimal The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abacuses are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal. The abacus was in use to the modern computer A computer is a programmable machine that receives input, stores and manipulates data//information, and provides output in a useful format, where research on the most efficient implementations of addition continues to this day.

Contents

Notation and terminology

The plus sign

Addition is written using the plus sign The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning "more" and "less", respectively "+" between the terms; that is, in infix notation Infix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on . It is not as simple to parse by computers as prefix notation ( e.g. + 2 2 ) or postfix notation ( e.g. 2 2 + ), but many programming languages use it due to its familiarity. The result is expressed with an equals sign The equality sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Welshman Robert Recorde. The equals sign is placed between the things stated to be exactly the same, as in an equation. It is the Unicode and ASCII character 003D. For example,

1 + 1 = 2 (verbally, "one plus one is equal to two")
2 + 2 = 4 (verbally, "two plus two is equal to four")
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:

Columnar addition: 5 + 12 = 17

The sum of a series A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, where as infinite sequences and series continue indefinitely of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example,

The numbers or the objects to be added in general addition are called 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 Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division). Some authors call the first addend the augend. In fact, during the Renaissance The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Florence in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historic era, but since the changes of the Renaissance were not uniform across Europe, this is a general use of the, many authors did not consider the first addend an "addend" at all. Today, due to the symmetry of addition, "augend" is rarely used, and both terms are generally called addends.[3]

All of this terminology derives from Latin Latin or sometimes Roman is an Italic language originally spoken in Latium and Ancient Rome. Although often considered a dead language, in view of the fact that it has no native, fluent speakers, Latin continues to be taught in schools and has been, and currently is, used in the process of new word production in modern languages from many. "Addition" and "add" are English English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into South-East Scotland under the influence of the Anglian medieval kingdom of Northumbria. Following the economic, political, military, scientific, cultural, and colonial influence of Great Britain and the United Kingdom from the 18th century, and of words derived from the Latin verb In syntax, a verb, from the Latin verbum meaning word, is a word that conveys action (bring, read, walk, run, murder), or a state of being (exist, stand). In most languages, verbs are inflected (modified in form) to encode tense, aspect, mood and voice. A verb may also agree with the person, gender, and/or number of some of its arguments, such as addere, which is in turn a compound In linguistics, a compound is a lexeme that consists of more than one stem. Compounding or composition is the word-formation that creates compound lexemes (the other word-formation process being derivation). Compounding or Word-compounding refers to the faculty and device of language to form new words by combining or putting together old words. In of ad "to" and dare "to give", from the Proto-Indo-European root The roots of the reconstructed Proto-Indo-European language are basic morphemes carrying a lexical meaning. By addition of suffixes, they form stems, and by addition of endings, these form grammatically inflected words (nouns or verbs) *deh₃- "to give"; thus to add is to give to.[3] Using the gerundive In linguistics, a gerundive is a particular verb form. The term is applied very differently to different languages; depending on the language, gerundives may be verbal adjectives, verbal adverbs, or finite verbs. Not every language has gerundives suffix An affix is a morpheme that is attached to a word stem to form a new word. Affixes may be derivational, like English -ness and pre-, or inflectional, like English plural -s and past tense -ed. They are bound morphemes by definition; prefixes and suffixes may be separable affixes. Affixation is, thus, the linguistic process speakers use to form new -nd results in "addend", "thing to be added".[4] Likewise from augere "to increase", one gets "augend", "thing to be increased".

Redrawn illustration from The Art of Nombryng, one of the first English arithmetic texts, in the 15th century[5]

"Sum" and "summand" derive from the Latin noun A noun can co-occur with an article or an attributive adjective. Verbs and adjectives can't. In the following, an asterisk in front of an example means that this example is ungrammatical 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 once common to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.[6] Addere and summare date back at least to Boethius Anicius Manlius Severinus Boëthius, commonly called Boethius was a Christian philosopher of the early 6th century. He was born in Rome to an ancient and important family which included emperors Petronius Maximus and Olybrius and many consuls. His father, Flavius Manlius Boethius, was consul in 487 after Odoacer deposed the last Western Roman, if not to earlier Roman writers such as Vitruvius Marcus Vitruvius Pollio was a Roman writer, architect and engineer (possibly praefectus fabrum during military service or praefect architectus armamentarius of the apparitor status group), active in the 1st century BC. By his own description Vitruvius served as a Ballista (artilleryman), the third class of arms in the military offices. He likely and Frontinus Sextus Julius Frontinus was one of the most distinguished Roman aristocrats of the late first century AD, but is best known to the post-Classical world as an author of technical treatises, especially one dealing with the aqueducts of Rome; Boethius also used several other terms for the addition operation. The later Middle English Middle English is the name given by historical linguists to the diverse forms of the English language in use between the late 11th century and about 1470, when the Chancery Standard, a form of London-based English, began to become widespread, a process aided by the introduction of the printing press into England by William Caxton in the late 1470s terms "adden" and "adding" were popularized by Chaucer Geoffrey Chaucer was an English author, poet, philosopher, bureaucrat, courtier and diplomat. Although he wrote many works, he is best remembered for his unfinished frame narrative The Canterbury Tales. Sometimes called the father of English literature, Chaucer is credited by some scholars as the first author to demonstrate the artistic legitimacy.[7]

Interpretations

Addition is used to model countless physical processes. Even for the simple case of adding natural numbers In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century, there are many possible interpretations and even more visual representations.

Combining sets

Possibly the most fundamental interpretation of addition lies in combining sets:

This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics; for the rigorous Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism. A religion, too, may be worn lightly, or applied with rigour definition it inspires, see Natural numbers below. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.[8]

One possible fix is to consider collections of objects that can be easily divided, such as pies The Proto-Indo-European language is the unattested, reconstructed common ancestor of the Indo-European languages, spoken by the Proto-Indo-Europeans. The existence of such a language has been accepted by linguists for over a century, and reconstruction is far advanced and quite detailed or, still better, segmented rods.[9] Rather than just 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:

A number-line visualization of the algebraic addition 2 + 4 = 6. A translation by 2 followed by a translation by 4 is the same as a translation by 6. A number-line visualization of the unary addition 2 + 4 = 6. A translation by 4 is equivalent to four translations by 1.

The sum a + b can be interpreted as a binary operation In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division 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 a + b play asymmetric roles, and the operation a + b is viewed as applying the unary operation In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a function +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

4 + 2 = 2 + 4 with blocks

Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result will be the same as the last one. 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". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law".

Associativity

2+(1+3) = (2+1)+3 with segmented rods

A somewhat subtler property of addition is associativity, which comes up when one tries to define repeated addition. Should the expression

"a + b + c"

be defined to mean (a + b) + c or a + (b + c)? That addition is associative tells us that the choice of definition is irrelevant. For any three numbers a, b, and c, it is true that

(a + b) + c = a + (b + c).

For example, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3). Not all operations are associative, so in expressions with other operations like subtraction, it is important to specify the order of operations.

Zero and one

5 + 0 = 5 with bags of dots

When adding zero to any number, the quantity does not change; zero is the identity element for addition, also known as the additive identity. In symbols, for any a,

a + 0 = 0 + a = a.

This law was first identified in Brahmagupta's Brahmasphutasiddhanta in 628, 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 wrote, "zero becomes the same as what is added to it", corresponding to the unary statement 0 + a = a. 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 a + 0 = a.[10]

In the context of integers, addition of one also plays a special role: for any integer a, the integer (a + 1) is the least integer greater than a, also known as the successor of a. Because of this succession, the value of some a + b can also be seen as the bth successor of a, making addition iterated succession.

Units

To numerically add physical quantities with units, they must first be expressed with common units. For example, 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.[11] A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants expect 1 + 1 to be 2, and they are comparatively surprised when a physical situation seems to imply that 1 + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.[12] Another 1992 experiment with older toddlers, between 18 to 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.[13]

Even some nonhuman animals show a limited ability to add, particularly primates. In a 1995 experiment imitating Wynn's 1992 result (but using eggplants instead of dolls), rhesus macaques and cottontop tamarins performed similarly to human infants. More dramatically, after being taught the meanings of the Arabic numerals 0 through 4, one chimpanzee was able to compute the sum of two numerals without further training.[14]

Elementary methods

Typically children master the art of counting first. When asked a problem requiring two items and three items to be combined, young children will model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they will 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.[15] 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 who is asked to add six and seven may know that 6+6=12 and then reason that 6+7 will be one more, or 13.[16] Such derived facts can be found very quickly and most elementary school children eventually rely on a mixture of memorized and derived facts to add fluently.[17]

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, but pattern-based strategies are more enlightening and, for most people, more efficient:[18]

As children grow older, they will commit more facts to memory, and learn to derive other facts rapidly and fluently. Many children never commit all the facts to memory, but can still find any basic fact quickly.[17]

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 ten, the extra digit is "carried" into the next column.[19] 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 other alternative methods.

Computers

Addition with an op-amp. See Summing amplifier for details.

Analog computers work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with an averaging lever. If the addends are the rotation speeds of two shafts, they can be added with a differential. A hydraulic adder can add the pressures in two chambers by exploiting Newton's second law to balance forces on an assembly of pistons. The most common situation for a general-purpose analog computer is to add two voltages (referenced to ground); this can be accomplished roughly with a resistor network, but a better design exploits an operational amplifier.[20]

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.

Part of Charles Babbage's Difference Engine including the addition and carry mechanisms

Adding machines, mechanical calculators whose primary function was addition, were the earliest automatic, digital computers. Wilhelm Schickard's 1623 Calculating Clock could add and subtract, but it was severely limited by an awkward carry mechanism. As he wrote to Johannes Kepler describing the novel device, "You would burst out laughing if you were present to see how it carries by itself from one column of tens to the next..." Adding 999,999 and 1 on Schickard's machine would require enough force to propagate the carries that the gears might be damaged, so he limited his machines to six digits, even though Kepler's work required more. By 1642 Blaise Pascal independently developed an adding machine with an ingenious gravity-assisted carry mechanism. Pascal's calculator was limited by its carry mechanism in a different sense: its wheels turned only one way, so it could add but not subtract, except by the method of complements. By 1674 Gottfried Leibniz made the first mechanical multiplier; it was still powered, if not motivated, by addition.[21]

"Full adder" logic circuit that adds two binary digits, A and B, along with a carry input Cin, producing the sum bit, S, and a carry output, Cout.

Adders execute integer addition in electronic digital computers, usually using binary arithmetic. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm taught to children. One slight improvement is the carry skip design, again following human intuition; one does not perform all the carries in computing 999 + 1, but one bypasses the group of 9s and skips to the answer.[22]

Since they compute digits one at a time, the above methods are too slow for most modern purposes. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all the floating-point operations as well as such basic tasks as address generation during memory 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. Almost all modern implementations are, in fact, hybrids of these last three designs.[23]

Unlike addition on paper, addition on a computer often changes the addends. On the ancient abacus and adding board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early Latin texts often claimed that in the process of adding "a number to a number", both numbers vanish.[24] In modern times, the ADD instruction of a microprocessor replaces the augend with the sum but preserves the addend.[25] In a high-level programming language, evaluating a + b does not change either a or b; to change the value of a one uses the addition assignment operator a += b.

Addition of natural and real 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.[26] (In mathematics education,[27] positive fractions are added before negative numbers are even considered; this is also the historical route.[28])

Natural numbers

Further information: Natural number

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 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:

Here, A U B 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, a mechanism that allows common elements to be separated out and therefore counted twice.

The other popular definition is recursive:

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 poset N².[31] 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.[32]

This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades.[33] He proved the associative and commutative properties, among others, through mathematical induction; for examples of such inductive proofs, see Addition of natural numbers.

Integers

Defining (−2) + 1 using only addition of positive numbers: (2 − 4) + (3 − 2) = 5 − 6. Further information: Integer

The simplest conception of an integer is that it consists of an absolute value (which is a natural number) and a sign (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:

Although this definition can be useful for concrete problems, it is far too complicated to produce elegant general proofs; there are too many cases to consider.

A much more convenient conception of the integers is the Grothendieck group construction. The essential observation is that every integer can be expressed (not uniquely) as the difference of two natural numbers, so we may as well define an integer as the difference of two natural numbers. Addition is then defined to be compatible with subtraction:

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:

The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic.[36] For a more rigorous and general discussion, see field of fractions.

Real numbers

Adding π2/6 and e using Dedekind cuts of rationals Further information: Construction of the 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:

This definition was first published, in a slightly modified form, by Richard Dedekind in 1872.[38] 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.[39]

Adding π2/6 and e using Cauchy sequences of rationals

Unfortunately, dealing with multiplication of Dedekind cuts is a case-by-case nightmare 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 a limit of a Cauchy sequence of rationals, lim an. Addition is defined term by term:

This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different.[41] 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.[42]

Generalizations

There are many things that can be added: numbers, vectors, matrices, spaces, shapes, sets, functions, equations, strings, chains... —Alexander Bogomolny

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

Addition in abstract algebra

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, in which vectors are interpreted as forces.

In modular arithmetic, 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 as the "exclusive or" function. In geometry, 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, which in turn generalizes to addition operations on many-dimensional tori.

The general theory of abstract algebra allows an "addition" operation to be any associative and commutative operation on a set. Basic algebraic structures with such an addition operation include commutative monoids and abelian groups.

Addition in set theory and category theory

A far-reaching generalization of addition of natural numbers is the addition of ordinal numbers and cardinal numbers in set theory. These give two different generalizations of addition of natural numbers to the transfinite. Unlike most addition operations, addition of ordinal numbers is not commutative. Addition of cardinal numbers, however, is a commutative operation closely related to the disjoint union operation.

In category theory, disjoint union is seen as a particular case of the coproduct operation, and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as Direct sum and Wedge sum, are named to evoke their connection with addition.

Related operations

Arithmetic

Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding x and subtracting x are inverse functions.

Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that is closed under subtraction.[43]

Multiplication can be thought of as repeated addition. If a single term x appears in a sum n times, then the sum is the product of n and x. If n is not a natural number, the product may still make sense; for example, multiplication by −1 yields the additive inverse of a number.

A circular slide rule

In the real and complex numbers, addition and multiplication can be interchanged by the exponential function:

ea + b = ea eb.[44]

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 groups, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated Lie algebra.[45]

There are even more generalizations of multiplication than addition.[46] 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 (1 + 1)(a + b) in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.[47]

Division is an arithmetic operation remotely related to addition. Since a/b = a(b−1), division is right distributive over addition: (a + b) / c = a / c + b / c.[48] However, division is not left distributive over addition; 1/ (2 + 2) is not the same as 1/2 + 1/2.

Ordering

Log-log plot of x + 1 and max (x, 1) from x = 0.001 to 1000[49]

The maximum operation "max (a, b)" is a binary operation similar to addition. In fact, if two nonnegative numbers a and b are of different orders of magnitude, then their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example in truncating Taylor series. However, it presents a perpetual difficulty in numerical analysis, essentially since "max" is not invertible. If b is much greater than a, then a straigh­tforward calculation of (a + b) − b can accumulate an unacceptable round-off error, perhaps even returning zero. See also Loss of significance.

The approximation becomes exact in a kind of infinite limit; if either a or b is an infinite cardinal number, their cardinal sum is exactly equal to the greater of the two.[50] Accordingly, there is no subtraction operation for infinite cardinals.[51]

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.[52] Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.[53]

Tying these observations together, tropical addition is approximately related to regular addition through the logarithm:

log (a + b) ≈ max (log a, log b),

which becomes more accurate as the base of the logarithm increases.[54] The approximation can be made exact by extracting a constant h, named by analogy with Planck's constant from quantum mechanics,[55] and taking the "classical limit" as h tends to zero:

In this sense, the maximum operation is a dequantized version of addition.[56]

Other ways to add

Incrementation, also known as the successor operation, is the addition of 1 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, which is zero.[57] An infinite summation is a delicate procedure known as a series.[58]

Counting a finite set is equivalent to summing 1 over the set.

Integration is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable manifold. Integration over a zero-dimensional manifold reduces to summation.

Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier, usually a real or complex number. Linear combinations are especially useful in contexts where straigh­tforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics.

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 literature

Notes

  1. ^ From Enderton (p.138): "...select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."
  2. ^ Devine et al. p.263
  3. ^ a b Schwartzman p.19
  4. ^ "Addend" is not a Latin word; in Latin it must be further conjugated, as in numerus addendus "the number to be added".
  5. ^ Karpinski pp.56–57, reproduced on p.104
  6. ^ Schwartzman (p.212) attributes adding upwards to the Greeks and Romans, saying it was about as common as adding downwards. On the other hand, Karpinski (p.103) writes that Leonard of Pisa "introduces the novelty of writing the sum above the addends"; it is unclear whether Karpinski is claiming this as an original invention or simply the introduction of the practice to Europe.
  7. ^ Karpinski pp.150–153
  8. ^ See this article for an example of the sophistication involved in adding with sets of "fractional cardinality".
  9. ^ Adding it up (p.73) compares adding measuring rods to adding sets of cats: "For example, inches can be subdivided into parts, which are hard to tell from the wholes, except that they are shorter; whereas it is painful to cats to divide them into parts, and it seriously changes their nature."
  10. ^ Kaplan pp.69–71
  11. ^ Wynn p.5
  12. ^ Wynn p.15
  13. ^ Wynn p.17
  14. ^ Wynn p.19
  15. ^ F. Smith p.130
  16. ^ Carpenter, Thomas; Fennema, Elizabeth; Franke, Megan Loef; Levi, Linda; Empson, Susan (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. ISBN 0-325-00137-5.
  17. ^ a b Henry, Valerie J.; Brown, Richard S. (2008). "First-grade basic facts: An investigation into teaching and learning of an accelerated, high-demand memorization standard". Journal for Research in Mathematics Education 39 (2): 153–183. doi:10.2307/30034895.
  18. ^ a b c d e f g Fosnot and Dolk p. 99
  19. ^ The word "carry" may be inappropriate for education; Van de Walle (p.211) calls it "obsolete and conceptually misleading", preferring the word "trade".
  20. ^ Truitt and Rogers pp.1;44–49 and pp.2;77–78
  21. ^ Williams pp.122–140
  22. ^ Flynn and Overman pp.2, 8
  23. ^ Flynn and Overman pp.1–9
  24. ^ Karpinski pp.102–103
  25. ^ The identity of the augend and addend varies with architecture. For ADD in x86 see Horowitz and Hill p.679; for ADD in 68k see p.767.
  26. ^ Enderton chapters 4 and 5, for example, follow this development.
  27. ^ California standards; see grades 2, 3, and 4.
  28. ^ Baez (p.37) explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"
  29. ^ Begle p.49, Johnson p.120, Devine et al. p.75
  30. ^ Enderton p.79
  31. ^ For a version that applies to any poset with the descending chain condition, see Bergman p.100.
  32. ^ Enderton (p.79) observes, "But we want one binary operation +, not all these little one-place functions."
  33. ^ Ferreirós p.223
  34. ^ K. Smith p.234, Sparks and Rees p.66
  35. ^ Enderton p.92
  36. ^ The verifications are carried out in Enderton p.104 and sketched for a general field of fractions over a commutative ring in Dummit and Foote p.263.
  37. ^ Enderton p.114
  38. ^ Ferreirós p.135; see section 6 of Stetigkeit und irrationale Zahlen.
  39. ^ The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see Enderton p.117 for details.
  40. ^ Textbook constructions are usually not so cavalier with the "lim" symbol; see Burrill (p. 138) for a more careful, drawn-out development of addition with Cauchy sequences.
  41. ^ Ferreirós p.128
  42. ^ Burrill p.140
  43. ^ The set still must be nonempty. Dummit and Foote (p.48) discuss this criterion written multiplicatively.
  44. ^ Rudin p.178
  45. ^ Lee p.526, Proposition 20.9
  46. ^ Linderholm (p.49) observes, "By multiplication, properly speaking, a mathematician may mean practically anything. By addition he may mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."
  47. ^ Dummit and Foote p.224. For this argument to work, one still must assume that addition is a group operation and that multiplication has an identity.
  48. ^ For an example of left and right distributivity, see Loday, especially p.15.
  49. ^ Compare Viro Figure 1 (p.2)
  50. ^ Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the Axiom of Choice.
  51. ^ Enderton p.164
  52. ^ Mikhalkin p.1
  53. ^ Akian et al. p.4
  54. ^ Mikhalkin p.2
  55. ^ Litvinov et al. p.3
  56. ^ Viro p.4
  57. ^ Martin p.49
  58. ^ Stewart p.8

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How does the addition of KCl result in a monophasic action potential?
Q. Also, what should the monophasic action potential after the KCl addition look like?
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A. The elevation of the extracellular potassium concentration K+is known to shorten action potential duration. >shortening of MAP is dependent mainly on the activation of K ATP channels. >So potassium efflux from the cell causes a marked shortening of the action potential duration .
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