In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the additive identity of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
that is equipped with the
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
of
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
is an
element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
0 from
elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
and
rings.
Elementary examples
* The additive identity familiar from
elementary mathematics is zero, denoted
0. For example,
*:
* In the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s N (if 0 is included), the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s Z, the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s Q, the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s R, and the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s C, the additive identity is 0. This says that for a
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
''n'' belonging to any of these sets,
*:
Formal definition
Let ''N'' be a
group that is closed under the
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
of
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
, denoted
+. An additive identity for ''N'', denoted ''e'', is an element in ''N'' such that for any element ''n'' in ''N'',
: ''e'' + ''n'' = ''n'' = ''n'' + ''e''.
Further examples
* In a
group, the additive identity is the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the group, is often denoted 0, and is unique (see below for proof).
* A
ring or
field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
1 if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
(proved below).
* In the ring M
''m''×''n''(''R'') of ''m'' by ''n''
matrices over a ring ''R'', the additive identity is the zero matrix,
denoted O or 0, and is the ''m'' by ''n'' matrix whose entries consist entirely of the identity element 0 in ''R''. For example, in the 2×2 matrices over the integers M
2(Z) the additive identity is
*:
*In the
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
, 0 is the additive identity.
*In the ring of
functions from R to R, the function
mapping every number to 0 is the additive identity.
*In the
additive group of
vectors in R
''n'', the origin or
zero vector is the additive identity.
Properties
The additive identity is unique in a group
Let (''G'', +) be a group and let 0 and 0' in ''G'' both denote additive identities, so for any ''g'' in ''G'',
: 0 + ''g'' = ''g'' = ''g'' + 0 and 0' + ''g'' = ''g'' = ''g'' + 0'.
It then follows from the above that
: = + 0 = 0' + = .
The additive identity annihilates ring elements
In a system with a multiplication operation that
distributes over addition, the additive identity is a multiplicative
absorbing element, meaning that for any ''s'' in ''S'', . This follows because:
:
The additive and multiplicative identities are different in a non-trivial ring
Let ''R'' be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let ''r'' be any element of ''R''. Then
: ''r'' = ''r'' × 1 = ''r'' × 0 = 0
proving that ''R'' is trivial, i.e. ''R'' = . The
contrapositive
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a stat ...
, that if ''R'' is non-trivial then 0 is not equal to 1, is therefore shown.
See also
*
0 (number)
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
*
Additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (op ...
*
Identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
*
Multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
References
Bibliography
*David S. Dummit, Richard M. Foote, ''Abstract Algebra'', Wiley (3rd ed.): 2003, .
External links
*{{PlanetMath , urlname=UniquenessOfAdditiveIdentityInARing2 , title=Uniqueness of additive identity in a ring , id=5676
Abstract algebra
Elementary algebra
Group theory
Ring theory
0 (number)