What are Properties of Integers?
Properties are rules that always hold true when performing operations on integers. These properties help us simplify calculations and solve problems more easily.
The Main Properties of Integers under Addition and Multiplication:
| Property | Meaning | Addition Example | Multiplication Example |
|---|---|---|---|
| Closure Property | Sum/product of two integers is always an integer | \((-7) + 4 = -3\) (integer) | \(5 \times (-3) = -15\) (integer) |
| Commutative Property | Changing order doesn't change the result | \((-3) + 5 = 5 + (-3) = 2\) | \((-4) \times 6 = 6 \times (-4) = -24\) |
| Associative Property | Changing grouping doesn't change the result | \([2 + (-3)] + 5 = 2 + [(-3) + 5] = 4\) | \([2 \times (-3)] \times 4 = 2 \times [(-3) \times 4] = -24\) |
| Distributive Property | Multiplication distributes over addition | \(a \times (b + c) = a \times b + a \times c\) | Example: \(-2 \times (3 + 4) = -2 \times 7 = -14\) or \((-2 \times 3) + (-2 \times 4) = -6 + (-8) = -14\) |
Identity and Inverse Elements:
| Identity | Definition | Example |
|---|---|---|
| Additive Identity | \(a + 0 = a = 0 + a\) (0 is additive identity) | \((-5) + 0 = -5\) |
| Multiplicative Identity | \(a \times 1 = a = 1 \times a\) (1 is multiplicative identity) | \((-7) \times 1 = -7\) |
| Additive Inverse | \(a + (-a) = 0\) (opposite number) | \(6 + (-6) = 0\) |
| Multiplicative Inverse | \(a \times \frac{1}{a} = 1\) (only for non-zero integers, but reciprocal may not be integer) | \(3 \times \frac{1}{3} = 1\) (but \(\frac{1}{3}\) is not an integer) |
Special Properties:
- Multiplication by Zero: \(a \times 0 = 0\) for any integer \(a\)
- Zero Property of Division: \(0 \div a = 0\) (where \(a \neq 0\)), but \(a \div 0\) is undefined