3 + 2 = 5 with apples, a popular choice in textbooks^[1]

Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection. Repeated addition of the number one is the most basic form of counting. Image File history File links Addition01. ... Image File history File links Addition01. ... Binomial name Borkh. ... An addition reaction, in chemistry, is in its simplest terms an organic reaction where two or more molecules combine to form a larger one. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... ‹ The template below is being considered for deletion. ... Counting is the mathematical action of repeatedly adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers...

Performing addition is one of the simplest numerical tasks, accessible to infants as young as five months and even some animals.

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, Image File history File links PlusCM128. ... The plus and minus signs (+ and −) are used to represent the notions of positive and negative as well as the operations of addition and subtraction. ... Infix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on (e. ... The equal sign, equals sign, or = is a mathematical symbol used to indicate equality. ...

1 + 1 = 2 (verbally, "one plus one equals two")

2 + 2 = 4

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: Image File history File links AdditionVertical. ...

• A column of numbers, with the last number in the column underlined, usually indicates that the numbers in the column are to be added, with the sum written below the underlined number.
• A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number.^[2] For example,
        3½ = 3 + ½ = 3.5.
  This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead.

The numbers or the objects to be added are generally 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. Some authors call the first addend the augend. In fact, during the Renaissance, 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] An underline is one or more horizontal lines immediately below a portion of text. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, multiplication is an elementary arithmetic operation. ... The Renaissance (French for rebirth, or Rinascimento in Italian), was a cultural movement in Italy (and in Europe in general) that began in the late Middle Ages, and spanned roughly the 14th through the 17th century. ...

All of this terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere, which is in turn a compound of ad "to" and dare "to give", from the Indo-European root do- "to give"; thus to add is to give to.^[4] Using the gerundive suffix -nd results in "addend", "thing to be added".^[5] Likewise from augere "to increase", one gets "augend", "thing to be increased". For other uses, see Latin (disambiguation). ... The English language is a West Germanic language that originates in England. ... It has been suggested that Verbal agreement be merged into this article or section. ... In linguistics, a compound is a lexeme (a word) that consists of more than one other lexeme. ... It has been suggested that this article or section be merged with Proto-Indo-European roots. ... Be sure to check the discussion page (and feel free to remove this tag if this article is updated). ... Look up affix in Wiktionary, the free dictionary. ...

Redrawn illustration from The Art of Nombryng, one of the first English arithmetic texts, in the 15th century^[6]

"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 once common to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.^[7] Addere and summare date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus; Boethius also used several other terms for the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer.^[8]
Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In linguistics, a noun or noun substantive is a lexical category which is defined in terms of how its members combine with other grammatical kinds of expressions. ... Boethius teaching his students (initial in a 1385 Italian manuscript of the Consolation of Philosophy). ... Marcus Vitruvius Pollio (born ca. ... Sextus Julius Frontinus (c. ... Middle English is the name given by historical linguistics to the diverse forms of the English language spoken between the Norman invasion of 1066 and the mid-to-late 15th century, when the Chancery Standard, a form of London-based English, began to become widespread, a process aided by the... Geoffrey Chaucer (c. ...

Interpretations

Addition is used to model countless physical processes. Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

Combining sets

Possibly the most fundamental interpretation of addition lies in combining sets: Image File history File links AdditionShapes. ...

• When two or more collections are combined into a single collection, the number of objects in the single collection is the sum of the number 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 Natural numbers below. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.^[9] For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. ...

One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods.^[10] 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. This article is about the baked good, for other uses see Pie (disambiguation). ...

Extending a length

A second interpretation of addition comes from extending an inital 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.

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 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 +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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, a unary operation is an operation with only one operand. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ...

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. Symbolically, if a and b are any two numbers, then Image File history File links AdditionComm01. ... Image File history File links AdditionComm01. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

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". In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

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 Image File history File links AdditionAsc. ... Image File history File links AdditionAsc. ... In mathematics, associativity is a property that a binary operation can have. ...

"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. In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. ...

Zero and one

5 + 0 = 5 with bags of dots

If one adds 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, Image File history File links AdditionZero. ... Image File history File links AdditionZero. ... For other senses of this word, see zero or 0. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... The additive identity of a number n is the number which, when added to n will yield n. ...

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.^[11] Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) was an Indian mathematician and astronomer. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic... Events Khusro II of Persia overthrown Pippin of Landen becomes Mayor of the Palace Brahmagupta writes the Brahmasphutasiddhanta Births Deaths Empress Suiko of Japan Theodelinda, queen of the Lombards Categories: 628 ... Here is a chronology of the main Indian mathematicians: BC Yajnavalkya, 1800 BC, the author of the altar mathematics of the Shatapatha Brahmana. ... Events Christian missionary Ansgar visits Birka, trade city of the Swedes. ... Mahavira was a 10th century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. ... Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician-astronomer. ...

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 b^th successor of a, making addition iterated succession. ‹ The template below is being considered for deletion. ...

Units

In order to numerically add physical quantities with units, they must first be expressed with common unit. 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. The former Weights and Measures office in Middlesex, England. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ...

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.^[12] 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.^[13] 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.^[14] In psychology, habituation is an example of non-associative learning in which there is a progressive diminution of behavioral response probability with repetition of a stimulus. ... “Baby” redirects here. ... Mickey Mouse is an Academy Award-winning comic animal cartoon character who has become an icon for The Walt Disney Company. ... Boy toddler Toddler is a common term for a a young child who is learning to walk or toddle,[1] generally considered to be the second stage of development after infancy and occurring predominantly during the ages of 12 to 36 months old. ... Regional competition level table tennis, showing table, net, and player getting ready to return the ball with a winning backhand topspin stroke. ...

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.^[15] Families 15, See classification A primate is any member of the biological order Primates, the group that contains all the species commonly related to the lemurs, monkeys, and apes, with the latter category including humans. ... “Aubergine” redirects here. ... Binomial name Macaca mulatta (Zimmermann, 1780) The Rhesus Macaque (Macaca mulatta), often called the Rhesus Monkey, is one of the best known species of Old World monkeys. ... Binomial name Saguinus oedipus (Linnaeus, 1758) The Cottontop Tamarin (Saguinus oedipus), also known as the Pinché Tamarin, is a small New World monkey weighing less than 1lb (0. ... For other uses, see Arabic numerals (disambiguation). ... Binomial name (Blumenbach, 1775) distribution of Common Chimpanzee. ...

Elementary methods

Typically children master the art of counting first, and this skill extends into a form of addition called "counting-on"; asked to find three plus two, children count two past three, saying "four, five", and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers, and some even invent it independently.^[16] Those who count to add also quickly learn to exploit the commutativity of addition by counting up from the larger number. Counting is the mathematical action of repeatedly adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers...

Decimal system

Single-digit addition table with various strategies colored: 0 in blue; 1,2 in light blue; (near) doubles in (light) green; making ten in red; 5,10 in gray.^[17]

The prerequisitive to addition in the decimal system is the internalization 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] Image File history File links AdditionTable. ... Image File history File links AdditionTable. ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus... In psychology, memory is an organisms ability to store, retain, and subsequently recall information. ... It has been suggested that Rote memory be merged into this article or section. ...

• One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, intuition.
• Zero: Since zero is the additive identity, adding zero is trivial. Nonetheless, some children 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 fortunately, children find them relatively easy to grasp. near-doubles...
• Five and ten...
• Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14.

In traditional mathematics, to add multidigit numbers, one typically aligns the addends vertically and adds 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] For a more detailed description of this algorithm, see Elementary arithmetic: Addition. 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 different standards-based mathematics methods, but many mathematics curricula such as TERC omit any instruction in traditional methods familiar to parents or mathematics professionals in favor of exploration of new methods. Intuition is an unconscious form of knowledge. ... This article does not cite any references or sources. ... In mathematics, multiplication is an elementary arithmetic operation. ... Traditional mathematics is the term used for the style of mathematics instruction used for a period in the 20th century before the appearance of reform mathematics based on NCTM standards, so it is best defined by contrast with the alternatives. ... Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ... Principles and Standards for School Mathematics is a document produced in 1989 by the National Council of Teachers of Mathematics [5] (NCTM) to set forth a national vision for precollege mathematics education in the US and Canada. ... Investigations in Number, Data, and Space is a complete K-5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts. ...

• Fraction: Addition
• Scientific notation: Operations
• Roman arithmetic: Addition

For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... Scientific notation, also known as standard form, is a notation for writing numbers that is often used by scientists and mathematicians to make it easier to write large and small numbers. ... In Rome, merchants used Roman numerals to perform basic arithmetic operations. ...

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] Image File history File links No higher resolution available. ... This article illustrates some typical applications of solid-state integrated circuit operational amplifiers. ... A page from the Bombardiers Information File (BIF) that describes the components and controls of the Norden bombsight. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... Levers can be used to exert a large force over a small distance at one end by exerting only a small force over a greater distance at the other. ... An axle is a central shaft for a rotating wheel or gear. ... In an automobile and other four-wheeled vehicles, a differential is a device, usually consisting of gears, for allowing each of the driving wheels to rotate at different speeds, while supplying equal torque to each of them. ... This article is about pressure in the physical sciences. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... For the use of the term in optics, see piston (optics). ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ... It has been suggested that Ground conductor be merged into this article or section. ... Resistor symbols (non-European) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... An electronic circuit is an electrical circuit that also contains active electronic devices such as transistors or vacuum tubes. ... A 741 operational amplifier in a TO-5 metal can package An operational amplifier, usually referred to as an op-amp for brevity, is a DC-coupled high-gain electronic voltage amplifier with Differential Inputs and, usually, a single output. ...

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. This article is about the machine. ...

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 machanism 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] Part of Charles Babbages Difference Engine assembled after his death by Babbages son, using parts found in his laboratory. ... Part of Charles Babbages Difference Engine assembled after his death by Babbages son, using parts found in his laboratory. ... Part of Babbages Difference engine, assembled after his death by Babbages son, using parts found in his laboratory. ... adding machine Older adding machine. ... Wilhelm Schickard Wilhelm Schickard (April 22, 1592 – October 23, 1635) was a German polymath who built the first computer in 1623. ... Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer and astrologer, and a key figure in the 17th century astronomical revolution. ... Blaise Pascal (pronounced ), (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... A Pascaline, signed by Pascal in 1652 Blaise Pascal invented the second mechanical calculator, called alternatively the Pascalina or the Arithmetique, in 1645, the first being that of Wilhelm Schickard in 1623. ... In mathematics and computing, the method of complements is a technique used to subtract one number from another using only addition of positive numbers. ... “Leibniz” redirects here. ...

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

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] Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In electronics, an adder is a device which will perform the addition, S, of two numbers. ... In electronics, an adder is a device which will perform the addition, S, of two numbers. ... The binary or base-two numeral system is a system for representing numbers in which a radix of two is used; that is, each digit in a binary numeral may have either of two different values. ...

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] A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... In computer science, a memory address is a unique identifier for a memory location at which a CPU or other device can store a piece of data for later retrieval. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In computer science, an instruction typically refers to a single operation of a processor within a computer architecture. ... In computer science, a parallel algorithm, as opposed to a traditional serial algorithm, is one which can be executed a piece at a time in many different processing devices, and then put back together again at the end to get the correct result. ... The Carry lookahead adder is one type of adder used in digital logic. ...

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. It has been suggested that Abax be merged into this article or section. ... For other uses, see Latin (disambiguation). ... A microprocessor is a programmable digital electronic component that incorporates the functions of a central processing unit (CPU) on a single semiconducting integrated circuit (IC). ... A high-level programming language is a programming language that, in comparison to low-level programming languages, may be more abstract, easier to use, or more portable across platforms. ...

Addition of natural and real numbers

In order 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]) In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Mathematics education is a term that refers both to the practice of teaching and learning mathematics, as well as to a field of scholarly research on this practice. ...

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: In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ...

• Let N(S) be the cardinality of a set S. Take two disjoint sets A and B, with N(A) = a and N(B) = b. Then a + b is defined as N(A U B).^[29]

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 which allows any common elements to be separated out and therefore counted twice. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

The other popular definition is recursive:

• Let n⁺ be the successor of n, that is the number following n in the natural numbers, so 0⁺=1, 1⁺=2. Define a + 0 = a. Define the general sum recursively by a + (b⁺) = (a + b)⁺. Hence 1+1=1+0⁺=(1+0)⁺=1⁺=2.^[30]

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] In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of the natural numbers. ... A Sierpinski triangle —a confined recursion of triangles to form a geometric lattice. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...

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. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... Addition of natural numbers is the most basic arithmetic operation. ...

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: Image File history File links GrothInt. ... Image File history File links GrothInt. ... The integers are commonly denoted by the above symbol. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... A negative number is a number that is less than zero, such as −3. ...

• 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 a + b = |a| + |b|. If a and b are both negative, define a + b = −(|a|+|b|). If a and b have different signs, define a + b to be the difference between |a| and |b|, with the sign of the term whose absolute value is larger.^[34]

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: In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way. ...

• Given two integers a − b and c − d, where a, b, c, and d are natural numbers, define (a − b) + (c − d) = (a + c) − (b + d).^[35]

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: In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ...

• Define    

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. In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ...

Real numbers

Adding π²/6 and e using Dedekind cuts of rationals

Further information: Construction of 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: Image File history File links AdditionRealDedekind. ... Image File history File links AdditionRealDedekind. ... In mathematics, there are a number of ways of defining the real number system as an ordered field. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ...

• Define ^[37]

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] Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... See also: Other events of 1872 List of years in science . ...

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 a_n. Addition is defined term by term: Image File history File links AdditionRealCauchy. ... Image File history File links AdditionRealCauchy. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...

• Define ^[40]

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] Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ...

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. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

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: Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. ...

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

This addition operation is central to classical mechanics, in which vectors are interpreted as forces. Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In physics, force is anything that can cause a massive body to accelerate. ...

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. Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as sets and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. ... Boolean logic is a complete system for logical operations. ... Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... This article is about angles in geometry. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In geometry, a torus (pl. ...

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. In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ...

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 set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

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. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In topology, the wedge sum is a one-point union of a family of topological spaces. ...

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. 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

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. In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, −1 is the integer greater than negative two (−2) and less than 0. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...

A circular slide rule

In the real and complex numbers, addition and multiplication can be interchanged by the exponential function: Coordinated Science Laboratory at the University of Illinois File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Coordinated Science Laboratory at the University of Illinois File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The exponential function is one of the most important functions in mathematics. ...

e^{a + b} = e^a e^b.^[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] Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying variables— to simplify and drastically speed up computation. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... A typical 10 inch student slide rule (Pickett N902-T simplex trig). ... Often in science, engineering, or other quantitative disciplines, it is necessary to make approximations with various degrees of precision. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

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] In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

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. In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...

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. Image File history File links XPlusOne. ... Image File history File links XPlusOne. ... A log-log plot of y=x (green), y=x^2 (blue), and y=x^3 (red). ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ... As the degree of the Taylor series rises, it approaches the correct function. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. ... Loss of significance is an undesirable effect in calculations using floating-point arithmetic. ...

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] Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

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] Tropical geometry is the study of geometry within a tropical semiring (also known as the min-plus algebra due to the definition of the semiring). ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

Tying these observations together, tropical addition is approximately related to regular addition through the logarithm: Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

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: A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Fig. ... The classical limit is the ability of a physical theory to approximate or recover classical mechanics when considered over special values of its parameters. ...

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. An increment is an increase, either of some fixed amount, for example added regularly, or of a variable amount. ... A successor function is the label in the literature for what is actually an operation. ... ‹ The template below is being considered for deletion. ...

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] For evaluation of sums in closed form see evaluating sums. ... In mathematics, the empty sum, or nullary sum, is the result of adding no numbers. ... For other senses of this word, see zero or 0. ... In mathematics, a series is often represented as the sum of a sequence of terms. ...

Counting a finite set is equivalent to summing 1 over the set. Counting is the mathematical action of repeatedly adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers...

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. The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... In mathematics, the word continuum sometimes denotes the real line. ... This article or section is in need of attention from an expert on the subject. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...

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. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... Please refer to Real vs. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In game theory a mixed strategy is a strategy which chooses randomly between possible moves. ... In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ... Game theory is often described as a branch of applied mathematics and economics that studies situations where multiple players make decisions in an attempt to maximize their returns. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ... A quantum state is any possible state in which a quantum mechanical system can be. ... Fig. ...

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 mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...

In literature

• In chapter 9 of Lewis Carroll's Through the Looking-Glass, the White Queen asks Alice, "And you do Addition? ... What's one and one and one and one and one and one and one and one and one and one?" Alice admits that she lost count, and the Red Queen declares, "She can't do Addition".
• In George Orwell's Nineteen Eighty-Four, the value of 2 + 2 is questioned; the State contends that if it declares 2 + 2 = 5, then it is so. See Two plus two make five for the history of this idea.

Charles Lutwidge Dodgson (Lewis Carroll) Charles Lutwidge Dodgson (IPA: ) (January 27, 1832 – January 14, 1898), better known by the pen name Lewis Carroll, was an English author, mathematician, logician, Anglican clergyman and photographer. ... Through the Looking-Glass, and What Alice Found There (1871) is a work of childrens literature by Lewis Carroll (Charles Lutwidge Dodgson), generally categorized as literary nonsense. ... Eric Arthur Blair (25 June 1903 [1] [2] – 21 January 1950), better known by the pen name George Orwell, was an English author and journalist. ... This article is about the Orwell novel. ... The phrase two plus two make five (or 2 + 2 = 5) is sometimes used as a succinct and vivid representation of an illogical statement, especially one made and maintained to suit an ideological agenda. ...

Notes

Ancient Rome was a civilization that grew from a small agricultural community founded on the Italian Peninsula circa the 9th century BC to a massive empire straddling the Mediterranean Sea. ... Portrait of Fibonacci, probably not authentic Leonardo of Pisa or Leonardo Pisano (Pisa, c. ... x86 or 80x86 is the generic name of a microprocessor architecture first developed and manufactured by Intel. ... The Motorola 680x0/0x0/m68k/68k/68K family of CISC microprocessor CPU chips were 32-bit from the start, and were the primary competition for the Intel x86 family of chips in personal computers of the 1980s and early 1990s. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

References