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3-10 The Real Numbers
Learn to determine if a number is
rational or irrational.
Pre-Algebra
3-10 The Real Numbers
Pre-Algebra
3-10 The Real Numbers
Biologists classify animals based on shared
characteristics. The gray lesser mouse lemur is
an animal, a mammal, a primate, and a lemur.
Animals
Mammals
Primates
Lemurs
Pre-Algebra
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.
3-10 The Real Numbers
Recall that rational numbers can be written as
fractions. Rational numbers can also be written
as decimals that either terminate or repeat.
3
4
= 3.8
5
Pre-Algebra
23
= 0.6
1.44 = 1.2
3-10 The Real Numbers
Irrational numbers can only be written as
decimals that do not terminate or repeat. If a
whole number is not a perfect square, then its
square root is an irrational number.
2 ≈1.4142135623730950488016…
Helpful Hint
A repeating decimal may not appear to repeat
on a calculator, because calculators show a
finite number of digits.
Pre-Algebra
3-10 The Real Numbers
The set of real numbers consists of the set of
rational numbers and the set of irrational numbers.
Real Numbers
Rational numbers
Integers
Whole
numbers
Pre-Algebra
Irrational numbers
3-10 The Real Numbers
Examples 1: Classifying Real Numbers
Write all names that apply to each number.
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 2
4
=
=2
2
whole, integer, rational, real
Pre-Algebra
3-10 The Real Numbers
Examples 2: Determining the Classification of All
Numbers
State if the number is rational, irrational,
or not a real number.
A.
15
15 is a whole number that
is not a perfect square.
irrational
B.
0 3
rational
Pre-Algebra
0 3
=0
3-10 The Real Numbers
Examples 2: Determining the Classification of All
Numbers
State if the number is rational, irrational,
or not a real number.
C.
–9
not a real number
D.
4 9
rational
Pre-Algebra
2 32 3
=
4 9
3-10 The Real Numbers
The Density Property of real numbers
states that between any two real numbers
is another real number. This property is
also true for rational numbers, but not for
whole numbers or integers. For instance,
there is no integer between –2 and –3.
Pre-Algebra
3-10 The Real Numbers
Examples 3: Applying the Density Property of Real
Numbers
2
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.
Find a real number between 3
2 5
3 5
5 5
1 2
3
+3
÷2 =6
÷2 =7÷2=3
3
1
2
3
4
3 5 3 5 13 5 3 5 4
32
3
2
1
A real number between 3
and 3
is 3 .
5
5
2
Pre-Algebra
3-10 The Real Numbers
Recall that rational numbers can be written as
fractions. Rational numbers can also be written
as decimals that either terminate or repeat.
3
4
= 3.8
5
Pre-Algebra
23
= 0.6
1.44 = 1.2
3-10 The Real Numbers
Irrational numbers can only be written as
decimals that do not terminate or repeat. If a
whole number is not a perfect square, then its
square root is an irrational number.
2 ≈1.4142135623730950488016…
Helpful Hint
A repeating decimal may not appear to repeat
on a calculator, because calculators show a
finite number of digits.
Pre-Algebra
3-10 The Real Numbers
The set of real numbers consists of the set of
rational numbers and the set of irrational numbers.
Real Numbers
Rational numbers
Integers
Whole
numbers
Pre-Algebra
Irrational numbers
3-10 The Real Numbers
Try This: Example 1
Write all names that apply to each number.
A.
9
9
=3
whole, integer, rational, real
B.
C.
Pre-Algebra
–35.9
–35.9 is a terminating decimal.
rational, real
81 3
81 3
9
=
=3
3
whole, integer, rational, real
3-10 The Real Numbers
Try This: Examples 2
State if the 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
not a number, so not a real number
Pre-Algebra
3-10 The Real Numbers
Try This: Examples 2
State if the number is rational, irrational,
or not a real number.
C.
–7
not a real number
D.
64
81
rational
Pre-Algebra
8 98 9
64
=
81
3-10 The Real Numbers
The Density Property of real numbers
states that between any two real numbers
is another real number. This property is
also true for rational numbers, but not for
whole numbers or integers. For instance,
there is no integer between –2 and –3.
Pre-Algebra
3-10 The Real Numbers
Try This: Example 3
3
4
Find a real number between 4
and 4 .
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 7
4 7
4
+4
÷2
1
2
7 7
=8
÷2
3
4
5
1 2
=9÷2=4
6
4 7 4 7 4 7 14 7 4 7 4 7
42
4
1
A real number between 4 3 and 4
is 4 .
7
2
7
Pre-Algebra