Additive Inverse
Pronunciation: /ˈæd ɪ tɪv ɪnˈvɜrs/ ?
An additive inverse is a property of real numbers. For an
arbitrary
real number a, the additive inverse of
a is ^{-}a.^{[1]} This is because
a + ^{-}a = 0, where 0
is the
additive identity
for real numbers.
Examples
The additive inverse of 5 is ^{-}5 since 5 + ^{-}5 = 0.
The additive inverse of ^{-}5 is 5 since ^{-}5 + 5 = 0.
The additive inverse of ^{-}3 is 3 since ^{-}3 + 3 = 0.
The additive inverse of 3 is ^{-}3 since 3 + ^{-}3 = 0.
The additive inverse of x is ^{-}x since x + ^{-}x = 0.
The additive inverse of ^{-}p is p since ^{-}p + p = 0.
Understanding Check
Click on the check box of the correct answer.
- What is the additive inverse of 8?
a a No, 8 + a ≠ 0. The additive inverse of 8 is ^{-}8.
^{-}a ^{-}a No, 8 + ^{-}a ≠ 0. The additive inverse of 8 is ^{-}8.
8 8 No, 8 + 8 ≠ 0. The additive inverse of 8 is ^{-}8.
^{-}8^{-}8 Yes! 8 + ^{-}8 = 0. The additive inverse of 8 is ^{-}8.
- What is the additive inverse of ^{-}4?
x x No, ^{-}4 + x ≠ 0. The additive inverse of ^{-}4 is 4.
^{-}x ^{-}x No, ^{-}4 + ^{-}x ≠ 0. The additive inverse of ^{-}4 is 4.
4 4 Yes! ^{-}4 + 4 = 0.
^{-}4^{-}4 No, ^{-}4 + ^{-}4 ≠ 0. The additive inverse of ^{-}4 is 4.
- What is the additive inverse of r?
r r No, r + r ≠ 0. The additive inverse of r is ^{-}r.
^{-}r ^{-}r Yes! r + ^{-}r = 0.
4 4 No, r + 4 ≠ 0. The additive inverse of r is ^{-}r.
^{-}4^{-}4 No, r + ^{-}4 ≠ 0. The additive inverse of r is ^{-}r.
Other Additive Inverses
The additive inverse is defined for a number of math entities, including
vectors,
and
matrix.
The additive inverse for vector <a,b> is
<^{-}a,^{-}b>. This is because
<a,b> + <^{-}a,^{-}b> = <0,0>.
<0,0> is the
additive identity
for vectors.
The additive inverse
for a matrix A is found by multiplying matrix
A by the scalar ^{-}1:
^{-}1·A. The result of this operation is a
zero matrix.
References
- additive inverse. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2009-12-24). http://www.merriam-webster.com/dictionary/additive inverse.
- Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pp 59-60. Thomson, Brooks/Cole, 2005.
More Information
- McAdams, David. Inverse. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Inverse of an Operation.
Cite this article as:
Additive Inverse. 2009-12-24. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/additiveinverse.html.
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Revision History
2009-12-24: Added "References" (
McAdams, David.)
2008-10-05: Expanded 'More Information' (
McAdams, David.)
2008-05-14: Added additive inverse of vectors and matrices. Corrected typo changing 'additive identity' to 'additive inverse' (
McAdams, David.)
2008-04-17: Initial version (
McAdams, David.)