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
- and so on!
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
Property
Example
a + b = b + a
2 + 6 = 6 + 2
ab = ba
4 × 2 = 2 × 4
Associative
Property
Example
(a + b) + c = a + ( b + c )
(1 + 6) + 3 = 1 + (6 + 3)
(ab)c = a(bc)
(4 × 2) × 5 = 4 × (2 × 5)
Distributive
Property
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
Property
Example
a+b is real
2 + 3 = 5 is real
a×b is real
6 × 2 = 12 is real
Adding 0 leaves the real number unchanged, likewise for multiplying by 1:
Identity
Property
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
Property
Example
a + (−a ) = 0
6 + (−6) = 0
Multiplicative Inverse
Property
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
Property
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
Property
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
7229, 7230, 7233, 7234, 3172, 3173, 7231, 7232, 2422, 2423