Inverse of an Operation

Pronunciation: /ɪnˈvɜrs/ ?

Given a set P, an element p ∈ P, 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.

Examples

Real numbers

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

The multiplicative inverse of a real number p 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:

MemberMultiplicative
Inverse
11
-1-1
i-i
-ii

References

  1. inverse. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-03-02). http://www.merriam-webster.com/dictionary/inverse.
  2. Gillet, J. A.. Elementary Algebra, pg 24. Henry Holt and Company, 1896. (Accessed: 2010-03-02). http://www.archive.org/stream/elementaryalgebr00gillrich#page/n43/mode/1up/search/inverse.
  3. Chrystal, G.. Introduction to Algebra for the use of Secondary, 3rd edition, pp 4-5. Adam and Charles Black, 1902. (Accessed: 2010-03-02). http://www.archive.org/stream/introductiontoal00chryuoft#page/4/mode/1up/search/inverse.
  4. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pp 30-31. Thomson, Brooks/Cole, 2005.

More Information

  • McAdams, David. Multiplicative 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=Multiplicative%20Inverse.
  • McAdams, David. Additive 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=Additive%20Inverse.
  • McAdams, David. Identity. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Identity.

Printed Resources

Cite this article as:


Inverse of an Operation. 2010-03-02. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/inverse.html.

Translations

Image Credits

Revision History


2010-03-02: Added "References" (McAdams, David.)
2008-09-19: Expanded 'More Information' (McAdams, David.)
2008-09-17: Revised 'More Information' (McAdams, David.)
2008-07-07: Corrected spelling (McAdams, David.)
2008-06-20: Clarified wording (McAdams, David.)
2008-03-25: Changed More Information to match current standard (McAdams, David.)
2008-02-18: Corrected spelling of quaternary (McAdams, David.)
2007-08-10: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License