mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...

is an element which, when added to any element in the set, yields . One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s (if 0 is included), the

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s 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 (for example, The set of all ...

s 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s 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 for ...

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

number A number is a mathematical object used to count, measure, and label. The most basic 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 can ...

belonging to any of these sets, *:

n+0 = n = 0+n.

Formal definition

Let be a group that is closed under the operation of

, denoted +. An additive identity for , denoted , is an element in such that for any element in , :

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 is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

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 is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

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 of -by- matrices over a ring , the additive identity is the zero matrix, denoted or , and is the -by- matrix whose entries consist entirely of the identity element 0 in . For example, in the 2×2 matrices over the integers the additive identity is *:

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

*In the quaternions, 0 is the additive identity. *In the ring of functions from , 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 * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

s in 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 id ...

is the additive identity.

Properties

The additive identity is unique in a group

Let be a group and let and in both denote additive identities, so for any in , :

0+g = g = g+0, \qquad 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 in , . 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 be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, i.e. 0 = 1. Let be any element of . Then :

r = r \times 1 = r \times 0 = 0

proving that is trivial, i.e. The

contrapositive In logic and mathematics, contraposition, or ''transposition'', refers to the inference of going from a Conditional sentence, conditional statement into its logically equivalent contrapositive, and an associated proof method known as . The contrap ...

, that if 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)