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1.1 Apply Properties of Real Numbers
Goal  Study properties of real numbers.
Your Notes
VOCABULARY
Opposite
The opposite, or additive inverse, of any number b is b.
Reciprocal
1
The reciprocal, or multiplicative inverse, of any nonzero number b is .
b
SUBSETS OF REAL NUMBERS
The real numbers consist of the _rational_ numbers and the _irrational_ numbers. Two
subsets of the rational numbers are the _whole numbers_ (0,1, 2, 3...) and the _integers_
(3, 2, 1, 0,1, 2, 3...).
Rational Numbers
Irrational Numbers

Can be written as quotients of
integers

Cannot be written as quotients of
integers

Can be written as decimals that
terminate or repeat

Cannot be written as decimals that
terminate or repeat
Example 1
Graph real numbers on a number line
Graph the real numbers

13
and 6 on a number line.
5
Solution
Note that  13 = _2.6_. Use a calculator to approximate 6 to the nearest tenth: 6 
5
_2.4_. So, graph  13 between _–3_ and _–2_ and graph 6 between _2_ and _3_.
5
Your Notes
PROPERTIES OF ADDITION AND MULTIPLICATION
Let a, b, and c be real numbers.
Property
Addition
_Closure_
a + b is a real number.
Multiplication
ab is a real number.
Commutative
a + b = _b + a_
ab = ba
Associative
(a + b) + c = a + (b + c)
(ab)c = _a(bc)_
Identity
a + 0 = a, _0 + a_ = a
a  1 = a, _1  a_ = a
Inverse
a + (a) = _0_
a
1
= 1, a  0
a
The following property involves both addition and multiplication.
Distributive
a(b + c) = _ab_ + _ac_
Example 2
Identify properties of real numbers
Identify the property that the statement illustrates.
a. (6  3)  2 = 6  (3  2) __ Associative_ property of _multiplication_
b. 21 + (21) = 0
_ Inverse_ property of _addition_
Checkpoint Complete the following exercises.
1. Graph the numbers  3.2, 3 , 5 ,  2,  1 .
2
2
2. Identify the property that 10(6 + 8) = 10(6) + 10(8) illustrates.
Distributive property
Your Notes
DEFINING SUBTRACTION AND DIVISION
Subtraction is defined as _adding the opposite_. The opposite, or _additive inverse_, of
any number b is b. If b is positive, then b is negative. If b is negative, then b is
positive.
a  b = a + (b)
Definition of subtraction
Division is defined as _multiplying by the reciprocal_. The reciprocal, or _multiplicative
1
inverse_, of any nonzero number b is .
b
1
ab=a . b0
Definition of division
b
Example 3
Use properties and definitions of operations
Show that 9 + (b  9) = b.
9 + (b  9)
= 9 + [b + ( 9 )]
Definition of subtraction
= 9 + [( 9 ) + b]
Commutative property of addition
= [9 + (9)] + b
_Associative_ property of addition
= _0_ + b
Inverse property of addition
= _b_
Identity property of addition
Checkpoint Use properties and definitions of operations to show that the
statement is true.
3. 5(a  5) = a
 1
5(a  5)  5 a  
 5
Definition of division
1 
 5  a 
5 
Commutative prop, of x
 1
  5  a
 5
=1  a
Associative prop, of x
=a
Identity prop, of x
Inverse prop, of x
Home work
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