Identities
What is an identity?
An identity is an equality that is true for all permissible values of the variables, not just for some.
Identity
a + b = b + a
This equality is true for ANY values of a and b:
- a = 2, b = 3: 2 + 3 = 3 + 2 ✓
- a = -5, b = 10: -5 + 10 = 10 + (-5) ✓
- a = 0, b = 0: 0 + 0 = 0 + 0 ✓
- a = 100, b = -100: it works! ✓
An identity has no "solutions" — it always works!
Equation
x + 3 = 7
This equality is true only when x = 4:
- x = 4: 4 + 3 = 7 ✓
- x = 5: 5 + 3 = 8 ≠ 7 ✗
- x = 0: 0 + 3 = 3 ≠ 7 ✗
- x = -1: -1 + 3 = 2 ≠ 7 ✗
An equation has specific solutions!
Examples of Identities
✓ These are identities:
| Identity | Name |
|---|---|
| a + b = b + a | Commutative property |
| (a + b) + c = a + (b + c) | Associative property |
| a · 1 = a | Multiplication by one |
| a + 0 = a | Addition of zero |
| a · 0 = 0 | Multiplication by zero |
| (a + b)² = a² + 2ab + b² | Square of a sum |
✗ These are NOT identities:
| Equality | Why it is not an identity |
|---|---|
| a + b = 5 | True only for some values of a, b |
| x² = 4 | True only when x = ±2 |
| 2x = x | True only when x = 0 |
| a + a = a | True only when a = 0 |
| x² = x | True only when x = 0 or x = 1 |
Permissible Values of Variables
Important!
Permissible values are the values for which the expression is meaningful (defined).
An identity must hold for all permissible values!
Example: a/a = 1
This is an identity, but only when a ≠ 0
- a = 5: 5/5 = 1 ✓
- a = -3: -3/(-3) = 1 ✓
- a = 0: 0/0 — undefined!
Permissible values: all numbers except 0
Example: √(x²) = |x|
This is an identity for all real x
- x = 3: √9 = 3 = |3| ✓
- x = -3: √9 = 3 = |-3| ✓
- x = 0: √0 = 0 = |0| ✓
Permissible values: all real numbers