Pre - Algebra
Bench Mark 7
Know Rational Numbers Are Either Terminating Or Repeating Decimals
Warm Up
Write as a decimal.
1. 1
2
(answer: 0.5)
2. 7
5
(answer: 1.4)
3. 9
4
(answer: 2.25)
4. 3
8
(answer: 0.375)
Pre - Algebra
Bench Mark 7
Know Rational Numbers Are Either Terminating Or Repeating Decimals
Vocabulary:
rational number
number that can be expressed as a ratio
of two integers
terminating decimal
number when expressed as a decimal has a
finite decimal expansion; also called a
regular number
repeating decimal
number when expressed as a decimal has
a ‘final’ set of digits which repeat an infinite
number of times; also called a recurring decimal
period
the number of recurring digits in a repeating
decimal; also called the repetend
vinculum
a horizontal line placed over the recurring digits in
a repeating decimal
Pre - Algebra
Bench Mark 7
Know Rational Numbers Are Either Terminating Or Repeating Decimals
It is obvious that terminating decimals are rational numbers.
For example,
75
3
625
5
6
3
0.75 =
=
0.625 =
=
and 0.6 =
=
100 4
1000 8
10
5
It is much less evident that repeating decimals
are also rational numbers.
For example,
1 = 0.1111,
9
1 = 0.1666
6
and 4 = 0.363636
11
So we can see that repeating decimals are also rational numbers.
Pre - Algebra
Bench Mark 7
Know Rational Numbers Are Either Terminating Or Repeating Decimals
The real challenge will be to write a fraction in simplest form
given a repeating decimal.
To illustrate, lets make up a repeating decimal such as 0.606060.
We will name our number a = 0.606060
and multiply it by a power of 10,
then subtract the original a and the new number
so that the repeating decimal parts
cancel each other in the subtraction.
a = .606060
and multiply it by a power of 10,
10a = 6.06060
The decimals do not line up so try multiplying by 100.
100a = 60.6060
The decimals line up and the repeating parts will cancel.
100a = 60.606060
- a = 0.606060
99a = 60 Solve for a
and reduce.
a = 60 = 20
99 33
So, 0.606060 = 20
33
Pre - Algebra
Bench Mark 7
Know Rational Numbers Are Either Terminating Or Repeating Decimals
Example
Find the period of the repeating part of 1 .
11
. 09 09 09
Divide.
11 1.0 0 0 0 0 0
- 99
1 00
- 99
10 0
- 99
1
We can see that 2 digits (0 and 9) are repeating so the period is 2.
Pre - Algebra
Bench Mark 7
Know Rational Numbers Are Either Terminating Or Repeating Decimals
Practice
a. Convert 0.4444 to a fraction
in simplest form.
4
(answer: 9 )
b. Find the period of
32
the repeating part of
.
33
(answer: 2 )
c. Write 13 as a decimal.
11
Classify the decimal as
repeating or terminating.
(answer: 1.1818;
repeating)
d. Convert 1.0555 to a fraction
in simplest form.
(answer: 19 )
18
Pre - Algebra
Bench Mark 7
Know Rational Numbers Are Either Terminating Or Repeating Decimals
Quiz
1. Write 7 as a decimal.
8
Classify the decimal as
repeating or terminating.
(answer: 0.875;
terminating)
2. Convert 1.5555 to a fraction
in simplest form.
(answer: 14 )
9
3. Find the period of
2
the repeating part of 3 .
4. Convert 0.98019801 to a
fraction in simplest form.
(answer: 1)
5. Find the period of
85
the repeating part of 101 .
(answer: 4)
(answer: 99 )
101