# 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:

 Commutative Example a + b = b + a 2 + 6 = 6 + 2 ab = ba 4 × 2 = 2 × 4 Associative Example (a + b) + c = a + ( b + c ) (1 + 6) + 3 = 1 + (6 + 3) (ab)c = a(bc) (4 × 2) × 5 = 4 × (2 × 5) Distributive Example a × (b + c) = ab + ac 3 × (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:

 Closure Example a+b is real 2 + 3 = 5 is real a×b is real 6 × 2 = 12 is real

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

 Identity Example a + 0 = a 6 + 0 = 6 a × 1 = a 6 × 1 = 6

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

 Additive Inverse Example a + (−a ) = 0 6 + (−6) = 0 Multiplicative Inverse Example a × (1/a) = 1 6 × (1/6) = 1 But not for 0 as 1/0 is undefined

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

 Zero Product Example If ab = 0 then a=0 or b=0, or both a × 0 = 0 × a = 0 5 × 0 = 0 × 5 = 0

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

 Negation Example −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