identity element

mathematics

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June 29, 2026

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In mathematics, an identity element is a special number within a set that, when combined with any other element in that set using a specific operation, leaves the other element unchanged.1

For example:
* In the set of all integers, the number 0 is the identity element for addition. Adding 0 to any integer does not change that integer (e.g., 5 + 0 = 5).12
* In the set of positive integers, the number 1 is the identity element for multiplication. Multiplying any positive integer by 1 does not change that integer (e.g., 7 × 1 = 7).1

These identity elements are crucial in various mathematical systems, such as group theory.3

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mathematics
The theory of equations
This element is called the identity element of the group. For every element a there is an element, written a⁻¹, with the property that a * a⁻¹ = e = a⁻¹ * a. The element a⁻¹ is called the inverse of a. For every a, b, and c in
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This element is called the identity element of the group. For every element a there is an element, written a⁻¹, with the property that a * a⁻¹ = e = a⁻¹ * a. The element a⁻¹ is called the inverse of a. For every a, b, and c in
Read More