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 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'' of ...

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 numbers c ...

0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and

rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

Elementary examples

* The additive identity familiar from elementary mathematics is zero, denoted 0. For example, *:

5 + 0 = 5 = 0 + 5.

* 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 n ...

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 language o ...

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 rati ...

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 form ...

s C, the additive identity is 0. This says that for a

''n'' belonging to any of these sets, *:

n + 0 = n = 0 + n.

Formal definition

Let ''N'' be a group that is closed under the operation of

, 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 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 (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 M₂(Z) the additive identity is *:

0 = \begin0 & 0 \\ 0 & 0\end

*In the quaternions, 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

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s in R^''n'', the origin or

zero vector In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive ident ...

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: :

\begin
             s \cdot 0 &= s \cdot (0 + 0) = s \cdot 0 + s \cdot 0 \\
 \Rightarrow s \cdot 0 &= s \cdot 0 - s \cdot  0 \\
 \Rightarrow s \cdot 0 &= 0.
\end

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 statem ...

, that if ''R'' is non-trivial then 0 is not equal to 1, is therefore shown.

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)