Properties of Real Numbers
The properties of real numbers
allow us to manipulate expressions
and equations and find the values
of a variable.
Number Classification
• Natural numbers are the counting numbers. • Whole numbers are natural numbers and zero. • Integers are whole numbers and their opposites.• Rational numbers can be written as a fraction.• Irrational numbers cannot be written as a
fraction.• All of these numbers are real numbers.
Number Classifications
Subsets of the Real Numbers
I - IrrationalZ - Integers
W - Whole
N - Natural
Q - Rational
Classify each number -1 real, rational, integer
real, rational, integer, whole, natural
real, irrational
real, rational
real, rational, integer, whole
real, rational
6
1
2
-2.2220
Properties of Real Numbers
Commutative Property• Think… commuting to work.• Deals with ORDER. It doesn’t matter
what order you ADD or MULTIPLY.
• a+b = b+a• 4 • 6 = 6 • 4
Properties of Real Numbers
Associative Property• Think…the people you associate
with, your group.• Deals with grouping when you
Add or Multiply.• Order does not change.
Properties of Real Numbers
Associative Property
• (a + b) + c = a + ( b + c)
• (nm)p = n(mp)
Properties of Real Numbers
Additive Identity Property• s + 0 = s
Multiplicative Identity Property • 1(b) = b
Distributive Property
• a(b + c) = ab + ac
• (r + s)9 = 9r + 9s
Properties of Real Numbers
• 5 = 5 + 0• 5(2x + 7) =
10x + 35• 8 • 7 = 7 • 8• 24(2) = 2(24)• (7 + 8) + 2 = 2 +
(7 + 8)
Additive Identity
Distributive
Commutative
Commutative
Commutative
Name the Property
Name the Property
• 7 + (8 + 2) = (7 + 8) + 2• 1 • v + -4 = v + -4
• (6 - 3a)b = 6b - 3ab• 4(a + b) = 4a + 4b
• Associative• Multiplicative
Identity• Distributive • Distributive
Properties of Real Numbers
Reflexive Property
• a + b = a + b
The same expression is written on both sides of the equal sign.
Properties of Real Numbers
• If a = b then b = a
• If 4 + 5 = 9 then 9 = 4 + 5
Symmetric Property
Properties of Real NumbersTransitive Property
• If a = b and b = c then a = c
• If 3(3) = 9 and 9 = 4 +5, then 3(3) = 4 + 5
Properties of Real Numbers
Substitution Property
• If a = b, then a can be replaced by b.
• a(3 + 2) = a(5)
Name the property• 5(4 + 6) = 20 + 30• 5(4 + 6) = 5(10)• 5(4 + 6) = 5(4 + 6)• If 5(4 + 6) = 5(10) then
5(10) = 5(4 + 6)• 5(4 + 6) = 5(6 + 4)• If 5(10) = 5(4 + 6) and
5(4 + 6) = 20 + 30 then 5(10) = 20 + 30
• Distributive• Substitution• Reflexive• Symmetric
• Commutative• Transitive
