Additive Inverse

Pronunciation: /ˈæd ɪ tɪv ɪnˈvɜrs/ Explain

The additive inverse of a number is the number that, when added to the original numbers, gives zero. 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.

Check Mark Understanding Check

Click on the check box of the correct answer.

  1. What is the additive inverse of 8?
    no answer aincorrect a No, 8 + a ≠ 0. The additive inverse of 8 is -8.
    no answer -aincorrect -a No, 8 + -a ≠ 0. The additive inverse of 8 is -8.
    no answer 8incorrect 8 No, 8 + 8 ≠ 0. The additive inverse of 8 is -8.
    no answer -8correct-8 Yes! 8 + -8 = 0. The additive inverse of 8 is -8.
  2. What is the additive inverse of -4?
    no answer xincorrect x No, -4 + x ≠ 0. The additive inverse of -4 is 4.
    no answer -xincorrect -x No, -4 + -x ≠ 0. The additive inverse of -4 is 4.
    no answer 4correct 4 Yes! -4 + 4 = 0.
    no answer -4incorrect-4 No, -4 + -4 ≠ 0. The additive inverse of -4 is 4.
  3. What is the additive inverse of r?
    no answer rincorrect r No, r + r ≠ 0. The additive inverse of r is -r.
    no answer -rcorrect -r Yes! r + -r = 0.
    no answer 4incorrect 4 No, r + 4 ≠ 0. The additive inverse of r is -r.
    no answer -4incorrect-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. additive inverse. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 8/6/2018. http://www.merriam-webster.com/dictionary/additive inverse.
  2. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 59-60. Thomson, Brooks/Cole. 2005. 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. 6/12/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/additiveinverse.html.

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Revision History

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.)

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