HOME

TheInfoList




In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the additive identity of a set that is equipped with the operation of
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
is an
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
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 A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
0
0
from
elementary mathematics 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
, but additive identities occur in other mathematical structures where addition is defined, such as in
groups A group is a number of people 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 identi ...
and
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery) A ring is a round band, usually of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...
.


Elementary examples

* The additive identity familiar from
elementary mathematics 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
is zero, denoted
0
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s N (if 0 is included), the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s Z, the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s Q, the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s R, and the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s C, the additive identity is 0. This says that for a
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
''n'' belonging to any of these sets, *:n + 0 = n = 0 + n.


Formal definition

Let ''N'' be a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
that is closed under the operation of
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

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''.


Further examples

* In a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, the additive identity is the
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of the group, is often denoted 0, and is unique (see below for proof). * A ring or
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
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, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
1
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 Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
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 In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion a ...
, 0 is the additive identity. *In the ring of
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s from R to R, the function
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
every number to 0 is the additive identity. *In the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
of
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
s in R''n'', the origin or
zero vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, 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 Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
, 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 A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

0 (number)
*
Additive inverse In mathematics, the additive inverse of a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be ...
*
Identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
*
Multiplicative identity In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...


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)