Identity
Pronunciation: /aɪˈdɛn.tɪ.ti/ Explain
An identity is a value and an
operation such that
applying
the value and operation does not change the result.
The
additive identity
for real numbers
is 0. This is because one can add
0 to any value, and the result is the same.
Example: 3 + 0 = 3.
The
multiplicative identity
for real numbers is 1. This is because one can multiply
any value by 1, and the result is the original value.
Example: 5 · 1 = 5.
Identity  Description 
a + 0 = a 
The additive identity for real numbers is the
number 0. This is because, for any real number
a, a +
0 = a. 
a · 1 =
a 
The multiplicative identity for real
numbers is 1. You can see that
a · 1 = a. 

The identity for a square matrix under multiplication is
a diagonal matrix where every diagonal value is
1. 
 An identity is an
equation
that is true for all values of the
variables
used in the equation. One example of an identity is the equation
sin(x)^{2}+cos(x)^{2}
= 1. This identity is always true for any value of
x. An identity can be used to
solve
equations or find the derivative of equations by transforming an equation into
a more useful form:
Equation  Description 
sin(x)^{2} + sin(x) =
1  cos(x)^{2} 
Equation to solve. 
sin(x)^{2} +
cos(x)^{2} + sin(x) = 1 
Add cos(x)^{2} to both sides of
the equations. 
1 + sin(x) = 1 
Substitute 1 for sin(x)^{2} + cos(x)^{2}. 
sin(x) = 0 
Subtract 1 from both sides. 
x ∈ {0, π, 2π, 3π, ...} 
Transform equation from trigonometric form to set form. 
Table 1: Using an identity to solve an equation. 
References
 McAdams, David E.. All Math Words Dictionary, identity. 2nd Classroom edition 201501084799968. pg 9394. Life is a Story Problem LLC. January 8, 2015. Buy the book
 Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pg 17. www.archive.org. International Textbook Company. January 1963. Last Accessed 8/7/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up. Buy the book
 Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 2930. Thomson, Brooks/Cole. 2005. Last Accessed 8/7/2018. Buy the book
 Mary Jane Sterling. Algebra II For Dummies. pg 12. For Dummies. June 19, 2006. Last Accessed 8/7/2018. Buy the book
More Information
 McAdams, David E.. Additive Identity. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2/11/2010. https://www.allmathwords.org/en/a/additiveidentity.html.
Cite this article as:
McAdams, David E. Identity. 4/23/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/i/identity.html.
Image Credits
Revision History
4/23/2019: Updated equations and expressions to new format. (
McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
McAdams, David E.)
2/11/2010: Added "References". (
McAdams, David E.)
8/9/2008: Added More Information, definition 2. (
McAdams, David E.)
7/12/2007: Initial version. (
McAdams, David E.)