Identity
Pronunciation: /aɪˈdɛntɪti/ ?
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 1
by any value, and the result is the same. 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
- identity. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=identity&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
- Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pg 17. International Textbook Company, January 1963. (Accessed: 2010-01-12). http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up.
- Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pp 29-30. Thomson, Brooks/Cole, 2005.
- Mary Jane Sterling. Algebra II For Dummies, pg 12. For Dummies, June 19, 2006. (Accessed: 2010-02-11).
More Information
- McAdams, David. Additive Identity. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-02-11. http://www.allmathwords.org/article.aspx?lang=en&id=Additive Identity.
- McAdams, David. Multiplicative Identity. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-02-11. http://www.allmathwords.org/article.aspx?lang=en&id=Multiplicative Identity.
- McAdams, David. Trigonometric Identities. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-02-11. http://www.allmathwords.org/article.aspx?lang=en&id=Trigonometric Identities.
Printed Resources
Cite this article as:
Identity. 2010-02-11. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/identity.html.
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Revision History
2010-02-11: Added "References" (
McAdams, David.)
2008-08-09: Added More Information, definition 2 (
McAdams, David.)
2007-07-12: Initial version (
McAdams, David.)