OpenStax-CNX module: m31331
1
Rational Numbers
∗
Rory Adams
Free High School Science Texts Project
Mark Horner
Heather Williams
This work is produced by OpenStax-CNX and licensed under the
†
Creative Commons Attribution License 3.0
1 Introduction
As described in the chapter on review of past work, a number is a way of representing quantity. The numbers
that will be used in high school are all real numbers, but there are many dierent ways of writing any single
real number.
This chapter describes rational numbers.
Khan Academy video on Integers and Rational Numbers
This media object is a Flash object. Please view or download it at
<http://www.youtube.com/v/kyu-IQ-gBIg&arel=0&hl=en_US&feature=player_embedded&version=3>
Figure 1
∗ Version
1.5: Apr 20, 2011 4:46 am -0500
† http://creativecommons.org/licenses/by/3.0/
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2 The Big Picture of Numbers
http://cnx.org/content/m31331/1.5/
Figure 2
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The term whole number does not have a consistent denition. Various authors use it in many dierent ways.
We use the following denitions:
• natural numbers are (1, 2, 3, ...)
• whole numbers are (0, 1, 2, 3, ...)
• integers are (... -3, -2, -1, 0, 1, 2, 3, ....)
3 Denition
The following numbers are all rational numbers.
10
,
1
−1
,
−3
21
,
7
10
,
20
−3
6
(1)
You can see that all denominators and all numerators are integers.
Denition 1: Rational Number
A rational number is any number which can be written as:
a
b
(2)
where a and b are integers and b 6= 0.
Only fractions which have a numerator and a denominator (that is not 0) that are integers
are rational numbers.
tip:
This means that all integers are rational numbers, because they can be written with a denominator of 1.
Therefore
√
2
,
7
π
20
(3)
are not examples of rational numbers, because in each case, either the numerator or the denominator is
not an integer.
A number may not be written as an integer divided by another integer, but may still be a rational
number. This is because the results may be expressed as an integer divided by an integer. The rule is, if a
number can be written as a fraction of integers, it is rational even if it can also be written in another way as
well. Here are two examples that might not look like rational numbers at rst glance but are because there
are equivalent forms that are expressed as an integer divided by another integer:
−1, 33
133
=
,
−3
300
−3
−300
=
= −100213
6, 39
639
(4)
3.1 Rational Numbers
1. If a is an integer, b is an integer and c is irrational, which of the following are rational numbers?
(i)
5
6
(ii)
a
3
(iii)
Table 1
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b
2
(iv)
1
c
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Click here for the solution1
2. If a1 is a rational number, which of the following are valid values for a?
(i) 1
(ii) −10
(iii)
√
2
(iv) 2, 1
Table 2
Click here for the solution2
4 Forms of Rational Numbers
All integers and fractions with integer numerators and denominators are rational numbers. There are two
more forms of rational numbers.
4.1 Investigation : Decimal Numbers
You can write the rational number
1.
2.
3.
4.
5.
1
2
as the decimal number 0,5. Write the following numbers as decimals:
1
4
1
10
2
5
1
100
2
3
Do the numbers after the decimal comma end or do they continue? If they continue, is there a repeating
pattern to the numbers?
You can write a rational number as a decimal number. Two types of decimal numbers can be written as
rational numbers:
4
1. decimal numbers that end or terminate, for example the fraction 10
can be written as 0,4.
2. decimal numbers that have a repeating pattern of numbers, for example the fraction 31 can be written
as 0, 3̇. The dot represents recurring 3's i.e., 0, 333... = 0, 3̇.
For example, the rational number 56 can be written in decimal notation as 0, 83̇ and similarly, the decimal
number 0,25 can be written as a rational number as 14 .
You can use a bar over the repeated numbers to indicate that the decimal is a repeating
decimal.
tip:
5 Converting Terminating Decimals into Rational Numbers
A decimal number has an integer part and a fractional part. For example 10, 589 has an integer part of 10
and a fractional part of 0, 589 because 10 + 0, 589 = 10, 589. The fractional part can be written as a rational
number, i.e. with a numerator and a denominator that are integers.
Each digit after the decimal point is a fraction with a denominator in increasing powers of ten. For
example:
•
•
1
10
1
100
1 See
2 See
is 0, 1
is 0, 01
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the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3N>
http://cnx.org/content/m31331/1.5/
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This means that:
10, 589
=
10 +
8
100
589
10 1000
10589
1000
5
10
=
=
+
+
9
1000
(5)
5.1 Fractions
1. Write the following as fractions:
(a) 0, 1
(b) 0, 12
(c) 0, 58
(d) 0, 2589
Table 3
Click here for the solution3
6 Converting Repeating Decimals into Rational Numbers
When the decimal is a repeating decimal, a bit more work is needed to write the fractional part of the
decimal number as a fraction. We will explain by means of an example.
If we wish to write 0, 3̇ in the form ab (where a and b are integers) then we would proceed as follows
x =
0, 33333...
10x =
9x =
multiply by 10 on both sides
(subtracting the second equation from the rst equation)
3, 33333...
3
3
9
x =
=
(6)
1
3
And another example would be to write 5, 4̇3̇2̇ as a rational fraction.
x
=
1000x
=
5, 432432432...
5432, 432432432...
multiply by 1000 on both sides
(7)
999x
=
x
=
5427
(subtracting the second equation from the rst equation)
5427
999
=
201
37
For the rst example, the decimal was multiplied by 10 and for the second example, the decimal was
multiplied by 1000. This is because for the rst example there was only one digit (i.e. 3) recurring, while
for the second example there were three digits (i.e. 432) recurring.
In general, if you have one digit recurring, then multiply by 10. If you have two digits recurring, then
multiply by 100. If you have three digits recurring, then multiply by 1000. Can you spot the pattern yet?
The number of zeros is the same as the number of recurring digits.
√ Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like
2 = 1, 4142135... cannot be written with an integer numerator and denominator, because they do not have
a pattern of recurring digits. However, when possible, you should try to use rational numbers or fractions
instead of decimals.
3 See
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http://cnx.org/content/m31331/1.5/
OpenStax-CNX module: m31331
6.1 Repeated Decimal Notation
1. Write the following using the repeated decimal notation:
a. 0, 11111111...
b. 0, 1212121212...
c. 0, 123123123123...
d. 0, 11414541454145...
Click here for the solution4
2. Write the following in decimal form, using the repeated decimal notation:
a. 23
3
b. 1 11
5
c. 4 6
d. 2 19
Click here for the solution5
3. Write the following decimals in fractional form:
a. 0, 6333̇
b. 5, 313131
c. 0, 999999̇
Click here for the solution6
7 Summary
1. Real numbers can be either rational or irrational.
2. A rational number is any number which can be written as ab where a and b are integers and b 6= 0
3. The following are rational numbers:
a. Fractions with both denominator and numerator as integers.
b. Integers.
c. Decimal numbers that end.
d. Decimal numbers that repeat.
8 End of Chapter Exercises
1. If a is an integer, b is an integer and c is irrational, which of the following are rational numbers?
a. 65
b. a3
c. 2b
d. 1c
Click here for the solution7
2. Write each decimal as a simple fraction:
a. 0, 5
b. 0, 12
c. 0, 6
d. 1, 59
4 See
5 See
6 See
7 See
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the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3n>
the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3Q>
the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3v>
http://cnx.org/content/m31331/1.5/
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OpenStax-CNX module: m31331
e. 12, 277̇
Click here for the solution8
3. Show that the decimal 3, 211̇8̇ is a rational number.
Click here for the solution9
4. Express 0, 78̇ as a fraction ab where a, b ∈ Z (show all working).
Click here for the solution10
8 See
9 See
10 See
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the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/l3G>
the le at <http://cnx.org/content/m31331/latest/http://www.fhsst.org/lOf>
http://cnx.org/content/m31331/1.5/
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