MI terminating Page 1 Thursday, June 28, 2001 2:07 PM
STRAND: Number
TOPIC:
Decimal numbers
Terminating and
recurring decimals
To convert a fraction to a decimal number we divide the numerator (top number) by the denominator (bottom number). We stop
the division when the decimal number either terminates (there is no remainder) or recurs (a pattern of digits begins repeating). In
this investigation we are going to examine the types of fractions that produce both terminating and recurring decimal numbers.
LITERACY TASK • Find the meanings of these key terms:
1. terminating:.......................................................................................................................................................................
2. recurring:...........................................................................................................................................................................
3. prime number: ..................................................................................................................................................................
4. prime factor:......................................................................................................................................................................
5. numerator:.........................................................................................................................................................................
6. denominator: .....................................................................................................................................................................
Whether or not the decimal representation of a fraction will terminate depends on the prime factors of the denominator.
1. Complete the table below which illustrates this fact:
Fraction
Prime factors of the
denominator
Decimal equivalent
Terminating or
recurring?
2 and 3
0.16
Recurring
2 and 5
0.1
Terminating
1
--2
1
--3
1
--4
1
--5
1
--6
1
--7
1
--8
1
--9
1
-----10
1
-----11
 John Wiley & Sons Australia, Ltd 2001
MI terminating Page 2 Thursday, June 28, 2001 2:07 PM
STRAND: Number
TOPIC:
Decimal numbers
INVESTIGATION: Terminating and
recurring decimals
Prime factors of the
denominator
Fraction
Decimal equivalent
Terminating or
recurring?
1
-----12
1
-----14
1
-----15
1
-----16
1
-----20
1
-----24
1
-----25
1
-----30
1
-----35
1
-----40
1
-----45
1
-----50
2. Look at the terminating decimal numbers in the table. What do you notice about the prime factors of these denominators?
3. Use this information to determine if the following fractions will produce terminating or recurring decimal numbers:
13
(a) --------
1
(e) -------63
400
The decimal representation of the fraction 1--- produces the recurring decimal number 0.142857. It is said to have a period of
7
100
4.
17
(b) --------
250
--------(c) 137
300
-----(d) 15
6 as the pattern repeats every six digits. State the period for each of the recurring decimal numbers listed in the table above.
5. Find the period for the following fractions with prime denominators:
1
(a) -----
13
1
(b) -----
17
1
(c) -----
19
6. It has been claimed that the decimal representation of the fraction --1- , where p is a prime number, will have a period of
p
length (p – 1). Is this correct?
 John Wiley & Sons Australia, Ltd 2001