# Real Number Properties

Real Numbers have properties!

### Example: Multiplying by zero

When we multiply a real number by zero we get zero:

- 5 × 0 = 0
- −7 × 0 = 0
- 0 × 0.0001 = 0
- etc!

It is called the "Zero Product Property", and is listed below.

## Properties

Here are the main properties of the Real Numbers

Real Numbers are Commutative, Associative and Distributive:

Commutativeexample

a + b = b + a2 + 6 = 6 + 2

ab = ba4 × 2 = 2 × 4

Associativeexample

(a + b) + c = a + ( b + c ) (1 + 6) + 3 = 1 + (6 + 3)

(ab)c = a(bc)(4 × 2) × 5 = 4 × (2 × 5)

Distributiveexample

a × (b + c) = ab + ac3 × (6+2) = 3 × 6 + 3 × 2

(b+c) × a = ba + ca(6+2) × 3 = 6 × 3 + 2 × 3

Real Numbers are closed (the result is also a real number) under addition and multiplication:

Closureexample

a+b is real2 + 3 = 5 is real

a×b is real6 × 2 = 12 is real

Adding zero leaves the real number unchanged, likewise for multiplying by 1:

Identityexample

a + 0 = a6 + 0 = 6

a × 1 = a6 × 1 = 6

For addition the inverse of a real number is its negative, and for multiplication the inverse is its reciprocal:

Additive Inverseexample

a + (−a ) = 06 + (−6) = 0

Multiplicative Inverseexample

a × (1/a) = 16 × (1/6) = 1

*But not for 0 as 1/0 is undefined*

Multiplying by zero gives zero (the Zero Product Property):

Zero Productexample

If ab = 0 then a=0 or b=0, or both

a × 0 = 0 × a = 05 × 0 = 0 × 5 = 0

Multiplying two negatives make a positive, and multiplying a negative and a positive makes a negative:

Negationexample

−1 × (−a) = −(−a) = a−1 × (−5) = −(−5) = 5

(−a)(−b) = ab(−3)(−6) = 3 × 6 = 18

(−a)(b) = (a)(−b) = −(ab)−3 × 6 = 3 × −6 = −18