Multiplicative Inverse

In a monoid or multiplicative group where the operation is a product ·, the multiplicative inverse of any element g is the element g^(-1) such that g·g^(-1)=g^(-1)·g=1, with 1 the identity element.

The multiplicative inverse of a nonzero number z is its reciprocal 1/z (zero is not invertible). For complex z=x+iy!=0,


The inverse of a nonzero real quaternion h=x+yi+vj+wk (where x,y,v,w are real numbers, and not all of them are zero) is its reciprocal


where alpha=x^2+y^2+v^2+w^2.

The multiplicative inverse of a nonsingular matrix is its matrix inverse.


To detect the multiplicative inverse of a given element in the multiplication table of finite multiplicative group, traverse the element's row until the identity element 1 is encountered, and then go up to the top row. In this way, it can be immediately determined that -i is the multiplicative inverse of i in the multiplicative group C_4 formed by all complex fourth roots of unity.

See also

Additive Inverse, Invertible Element, Multiplicative Identity, Multiplicative Group

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Multiplicative Inverse." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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