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