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.
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:
Member  Multiplicative Inverse 
1  1 
1  1 
i  i 
i  i 

References
 inverse. merriamwebster.com. Encyclopedia Britannica. (Accessed: 20100302). http://www.merriamwebster.com/dictionary/inverse.
 Gillet, J. A.. Elementary Algebra, pg 24. Henry Holt and Company, 1896. (Accessed: 20100302). http://www.archive.org/stream/elementaryalgebr00gillrich#page/n43/mode/1up/search/inverse.
 Chrystal, G.. Introduction to Algebra for the use of Secondary, 3rd edition, pp 45. Adam and Charles Black, 1902. (Accessed: 20100302). http://www.archive.org/stream/introductiontoal00chryuoft#page/4/mode/1up/search/inverse.
 Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pp 3031. Thomson, Brooks/Cole, 2005.
More Information
 McAdams, David. Multiplicative Inverse. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 20090312. 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. 20090312. 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. 20090312. http://www.allmathwords.org/article.aspx?lang=en&id=Identity.
Printed Resources
Cite this article as:
Inverse of an Operation. 20100302. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/inverse.html.
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20100302: Added "References" (
McAdams, David.)
20080919: Expanded 'More Information' (
McAdams, David.)
20080917: Revised 'More Information' (
McAdams, David.)
20080707: Corrected spelling (
McAdams, David.)
20080620: Clarified wording (
McAdams, David.)
20080325: Changed More Information to match current standard (
McAdams, David.)
20080218: Corrected spelling of quaternary (
McAdams, David.)
20070810: Initial version (
McAdams, David.)