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

Addition is the mathematical process of putting things together. The apple is the pomaceous Fruit of the apple tree Species Malus domestica in the Rose family Rosaceae. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The plus sign "+" means that two numbers are added together. The plus and minus signs ( + and &minus) are Mathematical symbols used to represent the notions of positive and negative as well as the operations A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. 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, since 3 + 2 = 5. Besides counts of fruit, addition can also represent combining other physical and abstract quantities using different kinds of numbers: negative numbers, fractions, irrational numbers, vectors, and more. A negative number is a Number that is less than zero, such as −2 In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction

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, children learn to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Primary education is the first stage of Compulsory education. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day. An abacus, also called a counting frame, is a calculating tool used primarily by Asians for performing arithmetic processes A computer is a Machine that manipulates data according to a list of instructions.

## Notation and terminology

Addition is written using the plus sign "+" between the terms; that is, in infix notation. The plus and minus signs ( + and &minus) are Mathematical symbols used to represent the notions of positive and negative as well as the operations 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 result is expressed with an equals sign. History The "=" symbol that is now universally accepted by mathematics for equality was first recorded by Welsh mathematician Robert Recorde in The For example,

1 + 1 = 2 (verbally, "one plus one equals two")
2 + 2 = 4 (verbally, "two plus two equals 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:

• 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. An underline, also called an underscore, is one or more horizontal lines immediately below a portion of Writing.
• A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object [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. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere 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 ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. "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. English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States For English usage of verbs see the wiki article English verbs. In Linguistics, a compound is a Lexeme (less precisely a Word) that consists of more than one stem. The roots of the reconstructed Proto-Indo-European language (PIE are basic Morphemes carrying a Lexical meaning [4] Using the gerundive suffix -nd results in "addend", "thing to be added". In Linguistics, a gerundive is a particular Verb form The term is applied very differently to different languages depending on the language gerundives may An affix is a Morpheme that is attached to a stem to form a word [5] 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[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. Anicius Manlius Severinus Boethius (480&ndash524 or 525 was a Christian philosopher of the 6th century Marcus Vitruvius Pollio (born c 80–70 BC died after c 15 BC was a Roman Writer, Architect and Engineer (possibly praefectus fabrum Sextus Julius Frontinus (ca 40-103 AD 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 The later Middle English terms "adden" and "adding" were popularized by Chaucer. Middle English is the name given by Historical linguistics to the diverse forms of the English language spoken between the Norman invasion of Geoffrey Chaucer (c 1343 – 25 October 1400? was an English author poet Philosopher, bureaucrat, courtier and Diplomat. [8]

## 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 (also called counting number) can mean either an element of the set (the positive Integers or an

### Combining sets

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

• 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. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. [9]

One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. A pie is a baked dish which is usually made of a Pastry dough shell that covers or completely contains a filling of various sweet or Savoury ingredients [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.

### Extending a length

A second interpretation of addition comes from extending an initial length by a given length:

• When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension.
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. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two 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. In Mathematics, a unary operation is an operation with only one Operand, i 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. 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 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. In Mathematics, commutativity is the ability to change the order of something without changing the end result Symbolically, if a and b are any two numbers, then

a + b = b + a.

The fact that addition is commutative is known as the "commutative law of addition". 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 Operands, 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. In Mathematics, associativity is a property that a Binary operation can have 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. In Algebra and Computer programming, when a number or expression is both preceded and followed by a Binary operation, a rule is required for which operation

### 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 Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics the additive identity of a set which is equipped with the operation of Addition is an element which when added to 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. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. The main work of Brahmagupta, Brahmasphuta-siddhanta (The Opening of the Universe, written in the year 628, contains some remarkably advanced ideas including Events By Place Europe Pippin of Landen becomes Mayor of the Palace in Austrasia. 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. The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to Modern times Events By Place Europe Earliest date of composition for the Historia Brittonum, attributed to Nennius, and known for Mahavira was a 9th century Indian Mathematician from Gulbarga who asserted that the Square root of a Negative number did 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. Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician [11]

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. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity 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

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. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand

### Innate ability

Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation: infants look longer at situations that are unexpected. See also Habit (psychology In Psychology, habituation is the psychological process in humans and animals in which there is a decrease in behavioral [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. Mickey Mouse is a comic animal Cartoon character who has become an icon for The Walt Disney Company. 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. Toddler is a common term for a young Child who is learning to walk or "toddle", generally considered to be the second stage of development after infancy [14]

Even some nonhuman animals show a limited ability to add, particularly primates. A primate is a member of the biological order Primates ( Latin: "prime first rank" the group that contains Lemurs the Aye-aye 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. The eggplant, aubergine, or brinjal ( Solanum melongena) is a plant of the family Solanaceae (also known as the nightshades The Rhesus Macaque ( Macaca mulatta) often called the Rhesus Monkey, is one of the best known species of Old World monkeys Adult males measure The Cottontop Tamarin ( Saguinus oedipus) also known as the Pinché Tamarin, is a small New World monkey weighing less than 1lb (0 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. The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system The Common Chimpanzee ( Pan troglodytes) also known as the Robust Chimpanzee, is a great ape. [15]

### 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. 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 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.

### 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". The decimal ( base ten or occasionally denary) Numeral system has ten as its base. One could memorize all the facts by rote, but pattern-based strategies are more enlightening and, for most people, more efficient:[18]

• One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, intuition. In Psychology, memory is an organism's ability to store retain and subsequently retrieve information Rote learning is a Learning technique which avoids understanding of a subject and instead focuses on memorization. Intuition is apparent ability to acquire knowledge without a clear inference or the use of reason
• 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. Abstract algebra has an unrelated term Word problem for groups.
• 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. Traditional mathematics (sometimes classical math education) is a term used to describe the predominant methods of Mathematics education in the United States 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. Elementary arithmetic is the most basic kind of Mathematics: it concerns the operations of Addition, Subtraction, Multiplication, and division 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. Principles and Standards for School Mathematics is a book produced by the National Council of Teachers of Mathematics (NCTM in 2000 to set forth a national vision for precollege

### Computers

Addition with an op-amp. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object Scientific notation, also sometimes known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be In Mathematics, Roman arithmetic is the use of Arithmetical operations on Roman numerals. See Summing amplifier for details. This article illustrates some typical applications of solid-state integrated circuit Operational amplifiers A simplified schematic notation is used and the reader is reminded that many

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. A computer is a Machine that manipulates data according to a list of instructions.

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

"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. In electronics an adder or summer is a Digital circuit that performs Addition of numbers

Adders execute integer addition in electronic digital computers, usually using binary arithmetic. In electronics an adder or summer is a Digital circuit that performs Addition of numbers The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. 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. In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. In Computer science, a memory address is an identifier for a memory location at which a Computer program or a hardware device can store a piece of data Computer data storage, often called storage or memory, refers to Computer components devices and recording media that retain digital In Computer science, an instruction is a single operation of a processor defined by an Instruction set architecture. In Computer science control flow (or alternatively flow of control refers to the order in which the individual statements, instructions or Function To increase speed, modern designs calculate digits in parallel; these schemes go by such names as carry select, carry lookahead, and the Ling pseudocarry. In Computer science, a parallel algorithm, as opposed to a traditional Sequential algorithm, is one which can be executed a piece at a time on many different processing A carry look-ahead adder is a type of adder used in Digital logic. 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. Paper is thin material mainly used for writing upon printing upon or packaging On the ancient abacus and adding board, both addends are destroyed, leaving only the sum. An abacus, also called a counting frame, is a calculating tool used primarily by Asians for performing arithmetic processes 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. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. [24] In modern times, the ADD instruction of a microprocessor replaces the augend with the sum but preserves the addend. A microprocessor incorporates most or all of the functions of a Central processing unit (CPU on a single Integrated [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. In computing a high-level programming language is a Programming language with strong abstraction from the details of the computer

## 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 Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways [26] (In mathematics education,[27] positive fractions are added before negative numbers are even considered; this is also the historical route. 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 [28])

### Natural numbers

Further information: Natural number

There are two popular ways to define the sum of two natural numbers a and b. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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:

• Let N(S) be the cardinality of a set S. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. 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. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets 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, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated

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. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural 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. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement [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. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that Addition of natural numbers is the most basic arithmetic binary 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 integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the absolute value (or modulus) 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 −2 The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:

• For an integer n, let |n| be its absolute value. Let a and b be integers. If either a or b is zero, treat it as an identity. If a and b are both positive, define 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. In Mathematics, the Grothendieck group construction in Abstract algebra constructs an Abelian group from a Commutative Monoid in the 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:

• Given two integers ab and cd, where a, b, c, and d are natural numbers, define (ab) + (cd) = (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:

• Define    $\frac ab + \frac cd = \frac{ad+bc}{bd}.$

The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the Least common multiple of the Denominators [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 or field of quotients

### 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. In Mathematics, a Dedekind cut, named after Richard Dedekind, in a Totally ordered set S is a partition of it into two non-empty In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Mathematics, especially in Order theory, the greatest element of a subset S of a Partially ordered set (poset is an element of S The sum of real numbers a and b is defined element by element:

• Define $a+b = \{q+r \mid q\in a, r\in b\}.$[37]

This definition was first published, in a slightly modified form, by Richard Dedekind in 1872. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important [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. In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence Addition is defined term by term:

• Define $\lim_na_n+\lim_nb_n = \lim_n(a_n+b_n).$[40]

This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. [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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

### 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. Linear algebra is the branch of Mathematics concerned with 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 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. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Physics, a force is whatever can cause an object with Mass 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. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers Musical set theory provides concepts for categorizing musical objects and describing their relationships 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. Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of In geometry, the sum of two angle measures is often taken to be their sum as real numbers modulo 2π. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called This amounts to an addition operation on the circle, which in turn generalizes to addition operations on many-dimensional tori. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar

The general theory of abstract algebra allows an "addition" operation to be any associative and commutative operation on a set. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result Basic algebraic structures with such an addition operation include commutative monoids and abelian groups. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

### 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. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. These give two different generalizations of addition of natural numbers to the transfinite. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. 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, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated

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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological Some coproducts, such as Direct sum and Wedge sum, are named to evoke their connection with addition. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Topology, the wedge sum (sometimes wedge product, though not to be confused with the Exterior product, which also shares this terminology is a "one-point

## Related operations

### Arithmetic

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, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Why is &minus1 times &minus1 equal to 1? Why is &minus1 multiplied by &minus1 equal to 1? More generally why is a negative times a negative a positive? There are two ways In mathematics the additive inverse, or opposite, of a Number n is the number that 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:

ea + b = ea eb. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) [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. Before Calculators were cheap and plentiful people would use mathematical tables &mdashlists of numbers showing the results of calculation with varying arguments&mdash to simplify In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce The slide rule, also known as a slipstick, is a mechanical Analog computer. 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. Orders of approximation have been used not only in Science, Engineering, and other quantitative disciplines to make Approximations with various degrees In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie [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 Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. 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. In Science and Engineering, a log-log graph or log-log plot is a two-dimensional graph of numerical data that uses Logarithmic scales on both 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. 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 This approximation is extremely useful in the applications of mathematics, for example in truncating Taylor series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives However, it presents a perpetual difficulty in numerical analysis, essentially since "max" is not invertible. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) 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. For the acrobatic movement roundoff see Roundoff. A round-off error, also called rounding error, is the difference between the See also Loss of significance. 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. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" [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. Tropical geometry is a relatively new area in Mathematics, which might loosely be described as a piece-wise linear or skeletonized version of Algebraic geometry In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is negative infinity. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced [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. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce [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:

$\max(a,b) = \lim_{h\to 0}h\log(e^{a/h}+e^{b/h}).$

In this sense, the maximum operation is a dequantized version of addition. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The classical limit is the ability of a physical theory to approximate or "recover" Classical mechanics when considered over special values of its parameters [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 The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict Subset of the recursive Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity

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. In Mathematics, the empty sum, or nullary sum, is the result of adding no numbers in Summation for example [57] An infinite summation is a delicate procedure known as a series. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with [58]

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

Integration is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable manifold. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, the word continuum has at least two distinct meanings outlined in the sections below A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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 Game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise this fully determines the player's behaviour In Game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise this fully determines the player's behaviour Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, Quantum superposition is the fundamental law of Quantum mechanics. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

Convolution is used to add two independent random variables defined by distribution functions. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable 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

• In chapter 9 of Lewis Carroll's Through the Looking-Glass, the White Queen asks Alice, "And you do Addition? . Charles Lutwidge Dodgson (ˈdɒdsən (27 January 1832 &ndash 14 January 1898 better known by the Pen name Lewis Carroll (/ˈkærəl/ was an English Through the Looking-Glass and What Alice Found There ( 1871) is a work of Children's literature by Lewis Carroll (Charles Lutwidge Dodgson . . 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. Eric Arthur Blair (25 June 1903 – 21 January 1950 who used the Pseudonym George Orwell, was an English writer Nineteen Eighty-Four (also titled 1984) by George Orwell (the pen name of Eric Arthur Blair) is a 1949 English Novel See Two plus two make five for the history of this idea. "2 + 2 = five" redirects here For information about the song by Radiohead see 2 + 2 = 5 (song The phrase "two plus two makes five"

## 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. ^ Schwartzman p. 19
4. ^ Schwartzman p. 19
5. ^ "Addend" is not a Latin word; in Latin it must be further conjugated, as in numerus addendus "the number to be added".
6. ^ Karpinski pp. 56–57, reproduced on p. 104
7. ^ Schwartzman (p. 212) attributes adding upwards to the Greeks and Romans, saying it was about as common as adding downwards. The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca Ancient Rome was a Civilization that grew out of a small agricultural community founded on the Italian Peninsula as early as the 10th century BC 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. Leonardo of Pisa (c 1170 – c 1250 also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or most commonly simply Fibonacci
8. ^ Karpinski pp. 150–153
9. ^ See this article for an example of the sophistication involved in adding with sets of "fractional cardinality".
10. ^ 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. "
11. ^ Kaplan pp. 69–71
12. ^ Wynn p. 5
13. ^ Wynn p. 15
14. ^ Wynn p. 17
15. ^ Wynn p. 19
16. ^ F. Smith p. 130
17. ^ Compare figures in Van de Walle pp. 160–164
18. ^ 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. See also X86 assembly language The generic term x86 refers to the most commercially successful Instruction set architecture in the history of Personal 679; for ADD in 68k see p. The Motorola 680x0 / m68k / 68k / 68K is a family of 32-bit CISC Microprocessor CPU chips and was the primary 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. The ascending chain condition (ACC and descending chain condition (DCC are finiteness properties satisfied by certain algebraic structures most importantly ideals 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. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.
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

## References

History
• Bunt, Jones, and Bedient (1976). The historical roots of elementary mathematics. Prentice-Hall. ISBN 0-13-389015-5.
• Ferreirós, José (1999). Labyrinth of thought: A history of set theory and its role in modern mathematics. Birkhäuser. ISBN 0-8176-5749-5.
• Kaplan, Robert (2000). The nothing that is: A natural history of zero. Oxford UP. ISBN 0-19-512842-7.
• Karpinski, Louis (1925). Louis Charles Karpinski ( 5 August 1878 &ndash 25 January 1956) was an American mathematician born in Rochester New York The history of arithmetic. Rand McNally. LCC QA21.K3. The Library of Congress Classification ( LCC) is a system of Library classification developed by the Library of Congress.
• Schwartzman, Steven (1994). The words of mathematics: An etymological dictionary of mathematical terms used in English. MAA. The Mathematical Association of America ( MAA) is a professional society that focuses on Mathematics accessible at the undergraduate level ISBN 0-88385-511-9.
• Williams, Michael (1985). A history of computing technology. Prentice-Hall. ISBN 0-13-389917-9.
Elementary mathematics
• Davison, Landau, McCracken, and Thompson (1999). Mathematics: Explorations & Applications, TE, Prentice Hall. ISBN 0-13-435817-1.
• F. Sparks and C. Rees (1979). A survey of basic mathematics. McGraw-Hill. ISBN 0-07-059902-5.
Education
• Begle, Edward (1975). The mathematics of the elementary school. McGraw-Hill. The McGraw-Hill Companies Inc, ( is a Publicly traded corporation headquartered in Rockefeller Center in New York City. ISBN 0-07-004325-6.
• California State Board of Education mathematics content standards Adopted December 1997, accessed December 2005.
• D. Devine, J. Olson, and M. Olson (1991). Elementary mathematics for teachers, 2e, Wiley. John Wiley & Sons Inc, also referred to as Wiley, is a global Publishing company that markets its products to professionals and consumers students and instructors ISBN 0-471-85947-8.
• National Research Council (2001). The National Research Council (NRC of the USA is the working arm of the United States National Academy of Sciences and the United States National Academy of Adding it up: Helping children learn mathematics. National Academy Press. The United States National Academies comprises four organizations the United States National Academy of Sciences (NAS the United States National Academy of Engineering ISBN 0-309-06995-5.
• Van de Walle, John (2004). Elementary and middle school mathematics: Teaching developmentally, 5e, Pearson. ISBN 0-205-38689-X.
Cognitive science
• Baroody and Tiilikainen (2003). "Two perspectives on addition development". The development of arithmetic concepts and skills: 75. ISBN 0-8058-3155-X.
• Fosnot and Dolk (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Heinemann. ISBN 0-325-00353-X.
• Weaver, J. Fred (1982). "Interpretations of number operations and symbolic representations of addition and subtraction". Addition and subtraction: A cognitive perspective: 60. ISBN 0-89859-171-6.
• Wynn, Karen (1998). "Numerical competence in infants". The development of mathematical skills: 3. ISBN 0-86377-816-X.
Mathematical exposition
• Bogomolny, Alexander (1996). Addition. Interactive Mathematics Miscellany and Puzzles (cut-the-knot. org). Retrieved on 3 February, 2006.
• Dunham, William (1994). The mathematical universe. Wiley. ISBN 0-471-53656-3.
• Johnson, Paul (1975). From sticks and stones: Personal adventures in mathematics. Science Research Associates. ISBN 0-574-19115-1.
• Linderholm, Carl (1971). Mathematics made difficult. Wolfe. ISBN 0-7234-0415-1.
• Smith, Frank (2002). The glass wall: Why mathematics can seem difficult. Teachers College Press. ISBN 0-8077-4242-2.
• Smith, Karl (1980). The nature of modern mathematics, 3e, Wadsworth. ISBN 0-8185-0352-1.
• Bergman, George (2005). An invitation to general algebra and universal constructions, 2. 3e, General Printing. ISBN 0-9655211-4-1.
• Burrill, Claude (1967). Foundations of real numbers. McGraw-Hill. LCC QA248.B95. The Library of Congress Classification ( LCC) is a system of Library classification developed by the Library of Congress.
• D. Dummit and R. Foote (1999). Abstract algebra, 2e, Wiley. ISBN 0-471-36857-1.
• Enderton, Herbert (1977). Elements of set theory. Academic Press. Academic Press ( London, New York and San Diego) was an Academic Book Publisher that is now part of Elsevier. ISBN 0-12-238440-7.
• Lee, John (2003). Introduction to smooth manifolds. Springer. ISBN 0-387-95448-1.
• Martin, John (2003). Introduction to languages and the theory of computation, 3e, McGraw-Hill. ISBN 0-07-232200-4.
• Rudin, Walter (1976). Principles of mathematical analysis, 3e, McGraw-Hill. ISBN 0-07-054235-X.
• Stewart, James (1999). Calculus: Early transcendentals, 4e, Brooks/Cole. ISBN 0-534-36298-2.
Mathematical research
Computing
• M. Flynn and S. Oberman (2001). Advanced computer arithmetic design. Wiley. ISBN 0-471-41209-0.
• P. Horowitz and W. Hill (2001). The art of electronics, 2e, Cambridge UP. ISBN 0-521-37095-7.
• Jackson, Albert (1960). Analog computation. McGraw-Hill. LCC QA76.4 J3. The Library of Congress Classification ( LCC) is a system of Library classification developed by the Library of Congress.
• T. Truitt and A. Rogers (1960). Basics of analog computers. John F. Rider. LCC QA76.4 T7. The Library of Congress Classification ( LCC) is a system of Library classification developed by the Library of Congress.