Additive identity
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In mathematics, the additive identity of a set that 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.
Elementary examples
- The additive identity familiar from elementary mathematics is zero, denoted 0. For example,
5
+
0
=
5
=
0
+
5.
{\displaystyle 5+0=5=0+5.}
![{\displaystyle 5+0=5=0+5.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94a1282862a8d54ebe8865918542961bfb242d53)
- In the natural numbers
N
{\displaystyle \mathbb {N} }
(if 0 is included), the integers
Z
,
{\displaystyle \mathbb {Z} ,}
the rational numbers
Q
,
{\displaystyle \mathbb {Q} ,}
the real numbers
R
,
{\displaystyle \mathbb {R} ,}
and the complex numbers
C
,
{\displaystyle \mathbb {C} ,}
the additive identity is 0. This says that for a number n belonging to any of these sets,
n
+
0
=
n
=
0
+
n
.
{\displaystyle n+0=n=0+n.}
![{\displaystyle n+0=n=0+n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edb29cd5292322df7bdbbaf71b8f474a09152785)
Formal definition
Let N be a group that is closed under the operation of addition, 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
.
{\displaystyle e+n=n=n+e.}
Further examples
- In a group, the additive identity is the identity element 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 Mm × 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
)
{\displaystyle \operatorname {M} _{2}(\mathbb {Z} )}
the additive identity is
0
=
{\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}
![{\displaystyle 0={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cab4359bab8a3b9c2b65bde9e0a3e47c7672752)
- In the quaternions, 0 is the additive identity.
- In the ring of functions from
R
→
R
{\displaystyle \mathbb {R} \to \mathbb {R} }
, the function mapping every number to 0 is the additive identity.
- In the additive group of vectors in
R
n
,
{\displaystyle \mathbb {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
,
0
′
+
g
=
g
=
g
+
0
′
.
{\displaystyle 0+g=g=g+0,\qquad 0'+g=g=g+0'.}
It then follows from the above that
0
′
=
0
′
+
0
=
0
′
+
0
=
0
.
{\displaystyle {\color {green}0'}={\color {green}0'}+0=0'+{\color {red}0}={\color {red}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, s · 0 = 0. This follows because:
s
⋅
0
=
s
⋅
(
0
+
0
)
=
s
⋅
0
+
s
⋅
0
⇒
s
⋅
0
=
s
⋅
0
−
s
⋅
0
⇒
s
⋅
0
=
0.
{\displaystyle {\begin{aligned}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{aligned}}}
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
{\displaystyle r=r\times 1=r\times 0=0}
proving that R is trivial, i.e. R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown.
See also
References
- ^ Weisstein, Eric W. "Additive Identity". mathworld.wolfram.com. Retrieved 2020-09-07.
Bibliography
- David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3rd ed.): 2003, ISBN 0-471-43334-9.
External links