The equals symbol states that two quantities have the same value in the situation being considered. The identity symbol states that two expressions are equal for every permitted value of their variable or variables.
This distinction is part of IB Mathematics: Analysis and Approaches SL 1.6: Simple deductive proof, and it can be tested on either Paper 1 or Paper 2.
| Symbol | Meaning in a proof | Example |
|---|---|---|
| The two sides have equal values, possibly only for particular variable values. | is true when or . |
| The two expressions are identical for every value in their common domain. | . |
To verify the identity in the table, expand the left-hand side:
Because this algebraic relationship is true for every real value of , the overall statement may be written
By contrast, consider
This is an equation to solve. It is true only when , giving or ; therefore, it is not an identity.
The phrase “permitted value” matters because identities are restricted to their common domain. For example, only for , since the left-hand expression is undefined at . An identity does not remove domain restrictions. During a derivation, ordinary equals signs can still connect expressions line by line because each adjacent pair has the same value.
A common misconception is that is always incorrect when a relationship is true for every value. The equals symbol remains mathematically valid, but communicates more precisely that the statement is an identity. Do not use merely to connect steps while solving an equation; use if you need to show that two equations are logically equivalent.
Exam technique: For a “show” or “show that” identity, begin with one side, transform it using valid algebraic steps, and arrive at the other side. Do not manipulate both sides simultaneously, and use in the final statement when the equality holds throughout the stated domain.
