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.

Check Mark Understanding Check

Click on the check box of the correct answer.

  1. What is the additive inverse of 8?
    Empty check box aRed check box indicating incorrect answer a No, 8 + a ≠ 0. The additive inverse of 8 is -8.
    Empty check box -aRed check box indicating incorrect answer -a No, 8 + -a ≠ 0. The additive inverse of 8 is -8.
    Empty check box 8Red check box indicating incorrect answer 8 No, 8 + 8 ≠ 0. The additive inverse of 8 is -8.
    Empty check box -8Green check box indicating correct answer-8 Yes! 8 + -8 = 0. The additive inverse of 8 is -8.
  2. What is the additive inverse of -4?
    Empty check box xRed check box indicating incorrect answer x No, -4 + x ≠ 0. The additive inverse of -4 is 4.
    Empty check box -xRed check box indicating incorrect answer -x No, -4 + -x ≠ 0. The additive inverse of -4 is 4.
    Empty check box 4Green check box indicating correct answer 4 Yes! -4 + 4 = 0.
    Empty check box -4Red check box indicating incorrect answer-4 No, -4 + -4 ≠ 0. The additive inverse of -4 is 4.
  3. What is the additive inverse of r?
    Empty check box rRed check box indicating incorrect answer r No, r + r ≠ 0. The additive inverse of r is -r.
    Empty check box -rGreen check box indicating correct answer -r Yes! r + -r = 0.
    Empty check box 4Red check box indicating incorrect answer 4 No, r + 4 ≠ 0. The additive inverse of r is -r.
    Empty check box -4Red check box indicating incorrect answer-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. (Accessed: 2009-12-24). 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.

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.)

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