Transcript Rational and Irrational Numbers

Making Sense of Rational and
Irrational Numbers
Objectives: Identify number sets.
Write decimals as fractions.
Write fractions as decimals.
The set of real numbers is all numbers that can
be written on a number line. It consists of the set
of rational numbers and the set of irrational
numbers.
Real Numbers
Rational numbers
Integers
Whole
numbers
Irrational numbers
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
Rational Numbers
Natural Numbers - Natural counting numbers.
1, 2, 3, 4 …
Whole Numbers - Natural counting numbers and zero.
0, 1, 2, 3 …
Integers - Whole numbers and their opposites.
… -3, -2, -1, 0, 1, 2, 3 …
Rational Numbers - Integers, fractions, and decimals.
Ex:
-0.76, -6/13, 0.08, 2/3
Biologists classify animals based on shared
characteristics. The horned lizard is an animal, a
reptile, a lizard, and a gecko. Rational Numbers
are classified this way as well!
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.
Venn Diagram: Naturals, Wholes, Integers, Rationals
Real Numbers
Rationals
6.7
5

9
0.8
Integers
11
Wholes
Naturals
1, 2, 3...
5
0
3
2
7
Name all the sets of numbers to which the given
number belongs. Circle the most specific set.
1)  5
Integers, Rationals
2
2)
3
Rationals
3) 16
Naturals , Wholes , Integers, Rationals
4) 0
Wholes , Integers, Rationals
5)  0.7 Rationals
Irrational numbers can be written only as
decimals that do not terminate or repeat. They
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.
Reminder
• Real numbers are all
the positive, negative,
fraction, and decimal
numbers you have
heard of.
• They are also called
Rational Numbers.
• IRRATIONAL
NUMBERS are
usually decimals that
do not terminate or
repeat. They go on
forever.
• Examples: π
2
3
Identify each root as rational or irrational.
1) 10
2)
irrational
25 rational
6)
62
7) 81
irrational
rational
3) 15 irrational
8)  16 rational
4)  49 rational
9)
5)
50 irrational
99
irrational
10) 121 rational
Decimal to Fraction: A skill
you will need for this unit!
• To change a decimal to a fraction, take the
place value and reduce!
• 0.5 means 5 tenths, so 5/10.
• Now reduce 5/10 = ½
• 0.5 = 1/2
Converting Fractions and Decimals
Fraction
Decimal
3
8
means 3  8
0 37 5
8 3.000
24
60
56
40
40
0
0.375
To change a fraction to a decimal, take
the top divided by the bottom, or
numerator divided by the denominator.
Complete the table.
Fraction
4
5
3
100
7
20
7
6
10
1
9
8
Decimal
0.8
0.03
0.35
6.7
9.125
Repeating Decimals
Fraction
1
3
means 1  3
0 3 33...
3 1.000
9
10
9
10
9
1
Decimal
0.3
0.33
Every rational number (fraction) either terminates
or repeats when written as a decimal.
Repeating Decimals
Fraction
5
11
means 5  11
0 4 54 54 ...
11 5.00000
44
60
55
50
44
60
55
50
44
6
Decimal
0.454
0.454
0.45
Repeating Decimals
Fraction
5
6
means 5  6
0 8 33...
6 5.000
48
20
18
20
18
2
Decimal
0.83
0.833
0.83