Additive Inverse

Pronunciation: /ˈæd.ɪ.tɪv ˈɪn.vɜrs/ Explain

The additive inverse of a number is the number that, when added to the original number, gives zero. The additive inverse of a is ^-a.^[3] 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.

1. 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.
2. 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.
3. 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

1. McAdams, David E.. All Math Words Dictionary, additive inverse. 2nd Classroom edition 20150108-4799968. pg 11. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. additive inverse. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 6/12/2018. http://www.merriam-webster.com/dictionary/additive inverse. Buy the book
3. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 59-60. Thomson, Brooks/Cole. 2005. Last Accessed 6/12/2018. Buy the book

More Information

• McAdams, David E.. Inverse. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/19/2018. http://www.allmathwords.org/en/i/inverse.html.
• Definition of Additive Inverse. www.mathsisfun.com. Math Is Fun. 6/18/2018. https://www.mathsisfun.com/definitions/additive-inverse.html.

Cite this article as:

McAdams, David E. Additive Inverse. 4/11/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/additiveinverse.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

4/11/2019: Changed equations and expressions to new format. (McAdams, David E.)
4/9/2019: Updated formatting of math objects. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/18/2018: Added More Information entry. (McAdams, David E.)
6/12/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
12/24/2009: Added "References". (McAdams, David E.)
10/5/2008: Expanded 'More Information'. (McAdams, David E.)
5/14/2008: Added additive inverse of vectors and matrices. Corrected typo changing 'additive identity' to 'additive inverse' (McAdams, David E.)
4/17/2008: Initial version. (McAdams, David E.)