Inverse of an Operation

Pronunciation: /ˈɪn.vɜrs ʌv ɑn ˌɒ.pərˈeɪ.ʃən/ Explain

Given a set P, an element pP, a binary operation * on set P, and an identity I for set P with respect to operation *, the inverse of p (p-1) with respect to operation * is the member of set P such that p * p-1 = I.


Real numbers

The additive inverse of a real number p is -p, since p + -p = p - p = 0.

The multiplicative inverse of any real number p, except zero, is 1 / p, since p · 1 / p = p / p = 1.

Mnemonic What is a mnemonic?

When an operation is performed on an identity and any other member of the set, the result is identical to the other member of the set.
Example: 5 + 0 = 5. The 5 on the left is identical to the 5 on the right.

Finding Inverses

1 -1 i -i
1 1 -1 i -i
-1 -1 1 -i i
i i -i -1 1
-i -i i 1 -1
Figure 1 - Multiplication table for quaternary

To find the inverse of a member of a set, start with a multiplication table. First, you need to find the identity for the operation. For what member when the operation is applied to it and any other member of set, the result is the other member of the set. In the multiplication table for the quaternary in figure 1, the values in the row for 1 (highlighted in green) are identical to the values at the top, so 1 is the multiplicative identity for the quaternary.

1 -1 i -i
1 1 -1 i -i
-1 -1 1 -i i
i i -i -1 1
-i -i i 1 -1
Figure 2 - Multiplication table for quaternary

Now find all results of the operation that match the identity. In the table in figure 2, These results are highlighted in red. This gives us this table of multiplicative inverses:



  1. McAdams, David E.. All Math Words Dictionary, inverse of an operation. 2nd Classroom edition 20150108-4799968. pg 101-102. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. inverse. Encyclopedia Britannica. Merriam-Webster. Last Accessed 8/7/2018. Buy the book
  3. Gillet, J. A.. Elementary Algebra. pg 24. Henry Holt and Company. 1896. Last Accessed 8/7/2018. Buy the book
  4. Chrystal, G.. Introduction to Algebra for the use of Secondary School and Technical Colleges. 3rd edition. pp 4-5. Adam and Charles Black. 1902. Last Accessed 8/7/2018. Buy the book
  5. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 30-31. Thomson, Brooks/Cole. 2005. Last Accessed 8/7/2018. Buy the book

More Information

  • McAdams, David E.. Multiplicative Inverse. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/12/2009.

Cite this article as:

McAdams, David E. Inverse of an Operation. 4/23/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

Revision History

4/23/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
3/2/2010: Added "References". (McAdams, David E.)
9/19/2008: Expanded 'More Information'. (McAdams, David E.)
9/17/2008: Revised 'More Information'. (McAdams, David E.)
7/7/2008: Corrected spelling. (McAdams, David E.)
6/20/2008: Clarified wording. (McAdams, David E.)
3/25/2008: Changed More Information to match current standard. (McAdams, David E.)
2/18/2008: Corrected spelling of quaternary. (McAdams, David E.)
8/10/2007: Initial version. (McAdams, David E.)

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