Download The Closure Properties The Commutative Properties

Math 90
1.7 "Properties of Real Numbers"
Objectives:
*
Closure, commutative, associative, distributive properties.
*
Identity elements.
*
Inverses for addition and multiplication.
The Closure Properties
The closure properties guarantee that the sum, di¤erence, product, or quotient (except for division by zero) of any
two real numbers is also a real number.
Closure Properties:
If a and b are real numbers, then
1: a + b is a real number.
2: a
Identity Elements:
2: ab is a real number
a
3: is a real number (b 6= 0)
b
b is a real number.
0 is the identity element for addition.
1 is the identity element for multiplication
Example 1: (Using the closure properties)
Assume that x = 12 and y =
x
a) 2
y
2. Show that each expression represents a real number by …nding the real-number answer.
2x
b)
c) y 2
3y
The Commutative Properties
The commutative properties guarantee that addition or multiplication of two real numbers can be done in either
order.
Commutative Properties:
If a and b are real numbers, then
1: a + b = b + a commutative property of addition
2: ab = ba commutative property of multiplication
Example 2: (Verifying the commutative properties)
Let x = 2, y =
3; and z = 1. Show that the two expressions have the same value.
a) (x + y) + z; y + (x + z)
b) (xy) z; x (yz)
Page: 1
c) x2 yz 2 ;
x2 y z 2
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
1.7
The Distributive Property
The distributive property shows how to multiply the sum of two numbers by a third number.
Distributive Property:
If a; b and c are real numbers, then a (b + c) = ab + ac
:
Example 3: (Using the distributive properties)
Use the distributive property to write each expression without parentheses. Simplify each result, if possible.
a) 2 (z
3)
b)
a (x + y)
c)
4 x2 + x
The Additive and Multiplicative Inverses
Additive and Multiplicative Inverses:
Because a + ( a) = 0 ; the numbers a and
Because a
1
a
= 1 (a 6= 0)
a are called negative or additive inverses:
the numbers a and
;
1
are called reciprocals or multiplicative inverses.
a
Example 4: (Using the identity properties and additive/multiplicative inverses)
Give the additive and the multiplicative inverse of each number, if possible.
a)
1
3
b) 0
c)
Page: 2
2
Notes by Bibiana Lopez