Additive identity
Encyclopedia
In mathematics
the additive identity of a set which is equipped with the operation
of addition
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
0
from elementary mathematics
, but additive identities occur in other mathematical structures where addition is defined, such as in groups
and rings
.
of addition
, denoted ]but]. An additive identity for N is any element e such that for any element n in N,
It follows from the above that
, meaning that for any s in S, s·0 = 0. This can be seen because:
proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
the additive identity of a set which is equipped with the operation
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
of addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . 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....
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
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
0
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
from elementary mathematics
Elementary mathematics
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry...
, but additive identities occur in other mathematical structures where addition is defined, such as in groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
and rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
.
Elementary examples
- The additive identity familiar from elementary mathematicsElementary mathematicsElementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry...
is zero, denoted 00 (number)0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
. For example,- 5 + 0 = 5 = 0 + 5.
- In the natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s N and all of its supersetSuperSetSuperSet Software was a group founded by friends and former Eyring Research Institute co-workers Drew Major, Dale Neibaur, Kyle Powell and later joined by Mark Hurst...
s (the integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s Z, the rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s Q, the real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s R, or the complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s C), the additive identity is 0. Thus for any one of these numberNumberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s n,- n + 0 = n = 0 + n. The additive identity is zero in an addition problem
Formal definition
Let N be a set which is closed under the operationOperation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
of addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . 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....
, denoted ]but]. An additive identity for N is any element e such that for any element n in N,
- e + n = n = n + e.
Further examples
- In a groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
the additive identity is the identity elementIdentity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
of the group, is often denoted 0, and is unique (see below for proof). - A ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
or fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
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 trivialTrivial (mathematics)In mathematics, the adjective trivial is frequently used for objects that have a very simple structure...
(proved below). - In the ring Mm×n(R) of m by n matricesMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
over a ring R, the additive identity is denoted 0 and is the m by n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2 by 2 matrices over the integers M2(Z) the additive identity is - In the quaternions, 0 is the additive identity.
- In the ring of functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s from R to R, the function mappingMap (mathematics)In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
every number to 0 is the additive identity. - In the additive groupAdditive groupAn additive group may refer to:*an abelian group, when it is written using the symbol + for its binary operation*a group scheme representing the underlying-additive-group functor...
of vectors in Rn, the origin or zero vector is the additive identity.
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 follows from the above that
- (0') = (0') + 0 = 0' + (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 elementAbsorbing element
In mathematics, an absorbing element is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element...
, meaning that for any s in S, s·0 = 0. This can be seen because:
- s·0 = s·(0 + 0) = s·0 + s·0 ⇒ s·0 = 0 (by cancellation).
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, or 0 = 1. Let r be any element of R. Then- r = r × 1 = r × 0 = 0,
proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.
See also
- 0 (number)0 (number)0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
- Additive inverseAdditive inverseIn mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
- Identity elementIdentity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
- Multiplicative identity