Download Ch 4-8 The Real Numbers

4-8 The Real Numbers
Evaluating Algebraic Expressions
California
Standards
NS1.4 Differentiate between rational
and irrational numbers.
4-8 The Real Numbers
Biologists classify animals based on shared
characteristics.
TheAlgebraic
horned lizard Expressions
is an animal, a
Evaluating
reptile, a lizard, and a gecko.
Animal
Reptile
Lizard
Gecko
You already know that some
numbers can be classified as
whole numbers, integers, or
rational numbers. The number
2 is a whole number, an
integer, and a rational number.
It is also a real number.
4-8 The Real Numbers
The set of real numbers is all numbers that can
be written on a number line. It consists of the set
Evaluating Algebraic Expressions
of rational numbers and the set of irrational
numbers.
Real Numbers
Rational numbers
Integers
Whole
numbers
Irrational numbers
4-8 The Real Numbers
Evaluating Algebraic Expressions
Recall that rational numbers can be written as
the quotient of two integers (a fraction) or as
either terminating or repeating decimals.
3
4
= 3.8
5
2
= 0.6
3
1.44 = 1.2
4-8 The Real Numbers
Irrational numbers can be written only as
decimals that do not terminate or repeat. They
Evaluating Algebraic Expressions
cannot be written as the quotient of two
integers. If a whole number is not a perfect
square, then its square root is an irrational
number. For example, 2 is not a perfect square,
so 2 is irrational.
Caution!
A repeating decimal may not appear to
repeat on a calculator, because
calculators show a finite number of digits.
4-8 The Real Numbers
Additional Example 1: Classifying Real Numbers
Write
all classifications
that apply
to each number.
Evaluating
Algebraic
Expressions
A.
5 is a whole number that is
not a perfect square.
irrational, real
5
B. –12.75 –12.75 is a terminating decimal.
rational, real
C.
16
2
16
4
=
=2
2
2
whole, integer, rational, real
4-8 The Real Numbers
Check It Out! Example 1
WriteEvaluating
all classifications
that applyExpressions
to each number.
Algebraic
A.
9
9
=3
whole, integer, rational, real
B.
C.
–35.9
–35.9 is a terminating decimal.
rational, real
81
81
9
=
=3
3
3
3
whole, integer, rational, real
4-8 The Real Numbers
Evaluating Algebraic Expressions
A fraction with a denominator of 0 is undefined
because you cannot divide by zero. So it is not a
number at all.
4-8 The Real Numbers
Additional Example 2: Determining the
Classification of All Numbers
Evaluating
Algebraic
Expressions
State
if each number
is rational,
irrational,
or not a real number.
A.
21
irrational
B.
0
3
rational
0
=0
3
4-8 The Real Numbers
Additional Example 2: Determining the
Classification of All Numbers
Evaluating
Algebraic
Expressions
State
if each number
is rational,
irrational,
or not a real number.
C. 4
0
not a real number
4-8 The Real Numbers
Check It Out! Example 2
Evaluating
Algebraic
Expressions
State
if each number
is rational,
irrational,
or not a real number.
A.
23
23 is a whole number that
is not a perfect square.
irrational
B.
9
0
undefined, so not a real number
4-8 The Real Numbers
Check It Out! Example 2
Evaluating
Algebraic
Expressions
State
if each number
is rational,
irrational,
or not a real number.
C.
64
81
rational
8
9
8
64
=
9
81
4-8 The Real Numbers
Evaluating
Algebraic
Expressions
The
Density Property
of real
numbers
states that between any two real
numbers is another real number. This
property is not true when you limit
yourself to whole numbers or integers.
For instance, there is no integer between
–2 and –3.
4-8 The Real Numbers
Additional Example 3: Applying the Density
Property of Real Numbers
Evaluating Algebraic2Expressions
3
Find a real number between 3
and 3 .
5
5
There are many solutions. One solution is halfway
between the two numbers. To find it, add the
numbers and divide by 2.
2
3
1
+3
÷2 =6 5 ÷2 =7÷2=3
5
5
2
5
3
2
1
A real number between 3
and 3
is 3 .
5
5
2
Check:
4
1
2
3
Use a graph.
3 3 5 3 5 13 5 35
4
32
3
4-8 The Real Numbers
Check It Out! Example 3
3
4
Find
a
real
number
between
4
and
4
.
Evaluating Algebraic Expressions
7
7
There are many solutions. One solution is halfway
between the two numbers. To find it, add the
numbers and divide by 2.
3
4
4
+4
7
7
÷2
7
=8
÷2
7
1
=9÷2=4
2
4
1
3
A real number between 4
and 4
is 4 .
7
2
7
Check:
5
1
6
2
3
4
Use a graph.
47 4 7 4 7 14 7 4 7 4 7
42