# Inverse element

element with an inverse with respect to a given mathematical operation; element that can 'undo' the effect of another given element

The inverse element is the opposite of a number or equation. The inverses of ${\displaystyle x}$ in addition is ${\displaystyle -x}$ (called the additive inverse), and ${\displaystyle {\tfrac {1}{x}}}$ or ${\displaystyle x^{-1}}$ in multiplication (called the multiplicative inverse, or reciprocal).[1][2]

In general, given an element g in a group S equipped with an operation ${\displaystyle \circ }$, an inverse element of g is an element g* such that ${\displaystyle g\circ g^{*}}$ and ${\displaystyle g^{*}\circ g}$ are both equal to the identity element (of the group). Other examples of inverse include inverse function and matrix inverse.

## Related pages

• Field (mathematics), a set equipped with additive inverses and multiplicative inverses (except for zero)

## References

1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-09-08.
2. "Reciprocal". www.mathsisfun.com. Retrieved 2020-09-08.