The operator that leaves a given operand unchanged.
The identity element is the value that, when combined with any number under a given operation, leaves that number exactly as it was. It is the "do nothing" value for that operation.
Each operation has its own identity element:
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For addition, the identity element is 0. Adding 0 to any number leaves it unchanged: 7 + 0 = 7.
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For multiplication, the identity element is 1. Multiplying any number by 1 leaves it unchanged: 7 × 1 = 7.
The identity element is what makes the inverse property possible. To "undo" an operation and return to the original number, we need a target to land on, and that target is always the identity element. For example, adding the additive inverse of 7 (which is –7) brings us back to 0, the identity element for addition.
Understanding the identity element helps students recognize patterns across operations and reason more clearly about why certain algebraic moves are valid.
When Do Students Learn About the Identity Element?
Students work with identity elements from their earliest arithmetic, even before the concept is named.
Grades K–2 – Adding Zero and Multiplying by One
Students discover that adding 0 or multiplying by 1 does not change a number, building direct experience with both identity elements.
Grades 3–5 – Identity Properties by Name
Students learn the additive identity property (a + 0 = a) and multiplicative identity property (a × 1 = a) as named rules, connecting them to their arithmetic experience.
Grades 6+ – Identity Elements in Algebra and Beyond
Students apply identity elements when solving equations, simplifying expressions, and reasoning about the structure of mathematical operations.
