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. 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.

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: -A. The result of this operation is a zero matrix.

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
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

• McAdams, David E.. Inverse. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/19/2018. https://www.allmathwords.org/en/i/inverse.html.

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

### 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.)