Properties of Real Numbers
We use properties of real numbers in manipulating algebraic expressions, given that our variables most often represent real numbers. Having a solid understanding of these properties is useful in developing fluency in algebraic processes.
Closure Properties of Real Numbers
1) The sum of any two real numbers is a real number.
(In other words, if a and b are real, then so is a + b.)
2) The product of any two real numbers is a real number.
(In other words, if a and b are real, then so is ab.)
Commutative Properties of Real Numbers
1) Let a and b be real numbers, then a + b = b + a.
2) Let a and b be real numbers, then a · b = b · a.
Associative Properties of Real Numbers
1) Let a, b, and c be real numbers, then (a + b) + c = a + (b + c).
2) Let a, b, and c be real numbers, then (a · b) · c = a ·(b · c).
Identity Properties of Real Numbers
1) There is a unique real number, 0, such that for all real numbers a, a + 0 = 0 + a = a. We say that 0 is the additive identity.
2) There is a unique real number, 1, such that for all real numbers a, a · 1 = 1 · a = a. We say that 1 is the multiplicative identity.
Inverse Properties of Real Numbers
1) For all real numbers a, there exists a unique real number, denoted -a, such that a + (-a) = 0. We say that –a is the additive inverse of a.
2) For all real numbers a, where a
0, there exists a unique real number, denoted
, such that
= 1. We say that
is the multiplicative inverse of a.
Multiplicative Property of Zero
For every real number a, a · 0 = 0 · a = 0.
Division Property of Zero
For every real number a, where a 
0, 0
a = 0.
Distributive Properties of Real Numbers
1) Let a, b, and c be real numbers, then a · (b + c) = a · b + a · c.
2) Let a, b, and c be real numbers, then (b + c) · a = b · a + c · a.