# Dictionary Definition

addend n : a number that is added to another
number (the augend)

# User Contributed Dictionary

#### Translations

term of addition

- Czech: sčítanec

### See also

# Extensive Definition

Addition is the mathematical process of
putting things together. The plus sign "+"
means that two numbers
are added together. 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.

As a mathematical
operation, addition follows several important patterns. It is
commutative,
meaning that order does not matter, and it is associative, meaning that
one can add more than two numbers (see Summation).
Repeated addition of 1 is the same
as counting; addition
of 0
does not change a number. Addition also obeys predictable rules
concerning related operations such as subtraction and multiplication. All of
these rules can be proven,
starting with the
addition of natural numbers and generalizing up through the
real
numbers and beyond. General binary
operations that continue these patterns are studied in abstract
algebra.

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.
Mechanical aids range from the ancient abacus to the modern computer, where research on the
most efficient implementations of addition continues to this
day.

## Notation and terminology

Addition is written using the plus sign "+" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,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.
- A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number. 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.

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.
Using the gerundive
suffix -nd
results in "addend", "thing to be added". Likewise from augere "to
increase", one gets "augend", "thing to be increased".

"Sum" and "summand" derive from the Latin
noun summa "the highest,
the top" and associated verb summare. This is appropriate not only
because the sum of two positive numbers is greater than either, but
because it was once common to add upward, contrary to the modern
practice of adding downward, so that a sum was literally higher
than the addends. Addere and summare date back at least to
Boethius, 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.

## 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.### 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. However, it is not obvious how one should extend
this version of addition to include fractional numbers or negative
numbers.

One possible fix is to consider collections of
objects that can be easily divided, such as pies or, still better, segmented
rods. Rather than 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.

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.

## Properties

### Commutativity

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- a + b = b + a.

### Associativity

A somewhat subtler property of addition is associativity, which comes up when one tries to define repeated addition. Should the expression- "a + b + c"

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

### Zero and one

When adding zero to any number, the quantity does not change; zero is the identity element for addition, also known as the additive identity. In symbols, for any a,- a + 0 = 0 + a = a.

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^ 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.## Performing addition

### Innate ability

Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation: infants look longer at situations that are unexpected. A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants expect 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. 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.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.

### 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. Those who count to add also quickly learn to exploit the commutativity of addition by counting up from the larger number.### Decimal system

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:- 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. 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.

### Computers

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.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.

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

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.

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.

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. In modern times, the ADD instruction of a microprocessor replaces
the augend with the sum but preserves the addend. In a
high-level programming language, evaluating a + b does not
change either a or b; to change the value of a one uses the
addition assignment operator a += b.

## Addition of natural and real numbers

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. (In mathematics education, positive fractions are added before negative numbers are even considered; this is also the historical route.)### Natural numbers

further 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:- 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).

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.

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

### Integers

further 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:- 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.

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:

- 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).

### 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.

### Real numbers

further 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:- Define a+b = \.

Unfortunately, dealing with multiplication of
Dedekind cuts is a case-by-case nightmare similar to the addition
of signed integers. Another approach is the metric completion of
the rational numbers. A real number is essentially defined to be
the a limit of a Cauchy
sequence of rationals, lim an. Addition is defined term by
term:

- Define \lim_na_n+\lim_nb_n = \lim_n(a_n+b_n).

## Generalizations

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

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

### Addition in abstract algebra

In linear algebra, a vector space is an algebraic structure that allows for adding any two vectors and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (a,b) is interpreted as a vector from the origin in the Euclidean plane to the point (a,b) in the plane. The sum of two vectors is obtained by adding their individual coordinates:- (a,b) + (c,d) = (a+c,b+d).

In modular
arithmetic, the set of integers modulo 12 has twelve elements;
it inherits an addition operation from the integers that is central
to musical
set theory. The set of integers modulo 2 has just two elements;
the addition operation it inherits is known in Boolean
logic as the "exclusive
or" function. In geometry, the sum of two
angle measures is often
taken to be their sum as real numbers modulo 2π. This amounts to an
addition operation on the circle, which in turn generalizes
to addition operations on many-dimensional tori.

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

### Addition in set theory and category theory

A far-reaching generalization of addition of natural numbers is the addition of ordinal numbers and cardinal numbers in set theory. These give two different generalizations of addition of natural numbers to the transfinite. Unlike most addition operations, addition of ordinal numbers is not commutative. Addition of cardinal numbers, however, is a commutative operation closely related to the disjoint union operation.In category
theory, disjoint union is seen as a particular case of the
coproduct operation,
and general coproducts are perhaps the most abstract of all the
generalizations of addition. Some coproducts, such as Direct sum and
Wedge
sum, are named to evoke their connection with addition.

## Related operations

### Arithmetic

Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding x and subtracting x are inverse functions.Given a set with an addition operation, one
cannot always define a corresponding subtraction operation on that
set; the set of natural numbers is a simple example. On the other
hand, a subtraction operation uniquely determines an addition
operation, an additive inverse operation, and an additive identity;
for this reason, an additive group can be described as a set that
is closed under subtraction.

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

- ea + b = ea eb.

There are even more generalizations of
multiplication than addition. In general, multiplication operations
always distribute
over addition; this requirement is formalized in the definition of
a ring.
In some contexts, such as the integers, distributivity over
addition and the existence of a multiplicative identity is enough
to uniquely determine the multiplication operation. The
distributive property also provides information about addition; by
expanding the product
(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.

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. However, division is not left distributive over
addition; 1/ (2 + 2) is not the same as 1/2 + 1/2.

### Ordering

The maximum operation "max (a, b)" is a binary operation similar to addition. In fact, if two nonnegative numbers a and b are of different orders of magnitude, then their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example in truncating Taylor series. However, it presents a perpetual difficulty in numerical analysis, essentially since "max" is not invertible. If b is much greater than a, then a straightforward calculation of (a + b) - b can accumulate an unacceptable round-off error, perhaps even returning zero. See also Loss of significance.The approximation becomes exact in a kind of
infinite limit;
if either a or b is an infinite cardinal number, their cardinal sum
is exactly equal to the greater of the two. Accordingly, there is
no subtraction operation for infinite cardinals.

Maximization is commutative and associative, like
addition. Furthermore, since addition preserves the ordering of
real numbers, addition distributes over "max" in the same way that
multiplication distributes over addition:

- a + max (b, c) = max (a + b, a + c'').

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

- log (a + b) ≈ max (log a, log b),

- \max(a,b) = \lim_h\log(e^+e^).

### Other ways to add

Incrementation, also known as the successor operation, is the addition of 1 to a number.Summation
describes the addition of arbitrarily many numbers, usually more
than just two. It includes the idea of the sum of a single number,
which is itself, and the empty sum,
which is zero. An
infinite summation is a delicate procedure known as a series.

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

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

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

Convolution is
used to add two independent random
variables defined by distribution
functions. Its usual definition combines integration,
subtraction, and multiplication. In general, convolution is useful
as a kind of domain-side addition; by contrast, vector addition is
a kind of range-side addition.

## In literature

- 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.

## Notes

## References

- The historical roots of elementary mathematics
- Labyrinth of thought: A history of set theory and its role in modern mathematics
- The nothing that is: A natural history of zero
- The history of arithmetic }}
- The words of mathematics: An etymological dictionary of mathematical terms used in English
- A history of computing technology
- Mathematics: Explorations & Applications
- A survey of basic mathematics
- The mathematics of the elementary school
- California State Board of Education mathematics content standards Adopted December 1997, accessed December 2005.
- Elementary mathematics for teachers
- Adding it up: Helping children learn mathematics ">http://www.nap.edu/books/0309069955/html/index.html}}
- Elementary and middle school mathematics: Teaching developmentally
- Young mathematicians at work: Constructing number sense, addition, and subtraction
- The mathematical universe
- From sticks and stones: Personal adventures in mathematics
- Mathematics made difficult
- The glass wall: Why mathematics can seem difficult
- The nature of modern mathematics
- An invitation to general algebra and universal constructions ">http://math.berkeley.edu/~gbergman/245/index.html}}
- Foundations of real numbers }}
- Abstract algebra
- Elements of set theory
- Introduction to smooth manifolds
- Introduction to languages and the theory of computation
- Principles of mathematical analysis
- Calculus: Early transcendentals
- Litvinov, Maslov, and Sobolevskii (1999). Idempotent mathematics and interval analysis. Reliable Computing, Kluwer.
- Viro, Oleg (2000). Dequantization of real algebraic geometry on logarithmic paper. (HTML) Plenary talk at 3rd ECM, Barcelona.
- Advanced computer arithmetic design
- The art of electronics
- Analog computation }}
- Basics of analog computers }}

addend in Arabic: جمع

addend in Belarusian (Tarashkevitsa):
Складаньне

addend in Catalan: Suma

addend in Czech: Sčítání

addend in Danish: Addition

addend in German: Addition

addend in Estonian: Liitmine

addend in Modern Greek (1453-): Άθροιση

addend in Spanish: Suma

addend in French: Addition

addend in Scottish Gaelic: Cur-ris

addend in Korean: 덧셈

addend in Croatian: Zbrajanje

addend in Ido: Adiciono

addend in Indonesian: Penjumlahan

addend in Icelandic: Samlagning

addend in Italian: Addizione

addend in Latin: Additio

addend in Lithuanian: Sudėtis

addend in Lojban: sumji

addend in Dutch: Optellen

addend in Japanese: 加法

addend in Norwegian: Addisjon

addend in Polish: Dodawanie

addend in Portuguese: Adição

addend in Russian: Сложение (математика)

addend in Simple English: Addition

addend in Slovenian: Vsota

addend in Serbian: Сабирање

addend in Finnish: Yhteenlasku

addend in Swedish: Addition

addend in Tagalog: Pagdaragdag

addend in Thai: การบวก

addend in Yiddish: צוגעבן

addend in Chinese: 加法