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 different ways of writing any single real number.
This chapter describes rational numbers.
The term whole number does not have a consistent definition. Various authors use
it in many different ways. We use the following definitions:
- natural numbers are (1, 2, 3, ...)
- whole numbers are (0, 1, 2, 3, ...)
- integers are (... -3, -2, -1, 0, 1, 2, 3, ....)
The following numbers are all rational numbers.
10
1
,
21
7
,
-
1
-
3
,
10
20
,
-
3
6
10
1
,
21
7
,
-
1
-
3
,
10
20
,
-
3
6
(1)
You can see that all denominators and all numerators are integers.
- Definition 1: Rational Number
A rational number is any number which can be written as:
a
b
a
b
(2)
where aa and bb are integers and b≠0b≠0.
Only fractions which have a numerator and a denominator (that is not 0) that are integers
are rational numbers.
This means that all integers are rational numbers, because they can be written with a denominator of 1.
Therefore
2
7
,
π
20
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 first glance but are because there are equivalent forms that are expressed as an
integer divided by another integer:
-
1
,
33
-
3
=
133
300
,
-
3
6
,
39
=
-
300
639
=
-
100
213
-
1
,
33
-
3
=
133
300
,
-
3
6
,
39
=
-
300
639
=
-
100
213
(4)
- If aa is an integer, bb is an integer and cc is irrational, which of the following are rational numbers?
Table 1| (i) 5656 | (ii) a3a3 | (iii) b2b2 | (iv) 1c1c |
- If a1a1 is a rational number, which of the following are valid values for aa?
Table 2| (i) 1 | (ii) -10-10 | (iii) 22 | (iv) 2,12,1 |
All integers and fractions with integer numerators and denominators are rational numbers. There are two more forms of rational numbers.
You can write the rational number
1212 as the decimal number 0,5. Write the following numbers as
decimals:
-
1
4
1
4
-
1
10
1
10
-
2
5
2
5
-
1
100
1
100
-
2
3
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:
- decimal numbers that end or terminate, for example the fraction 410410 can be written as 0,4.
- decimal numbers that have a repeating pattern of numbers, for example the fraction 1313 can be written as
0,3˙0,3˙.
The dot represents recurring
33's i.e.,
0,333... = 0,3˙0,333... = 0,3˙.
For example, the rational number 5656 can be written in decimal notation as 0,83˙0,83˙ and similarly, the decimal number 0,25 can be written as a rational number as 1414.
You can use a bar over the repeated numbers to indicate that the decimal is a repeating decimal.
A decimal number has an integer part and a fractional part. For example 10,58910,589 has an integer part of 10 and a fractional part of 0,5890,589 because 10+0,589=10,58910+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:
- 110110 is 0,10,1
- 11001100 is 0,010,01
This means that:
10
,
589
=
10
+
5
10
+
8
100
+
9
1000
=
10
589
1000
=
10589
1000
10
,
589
=
10
+
5
10
+
8
100
+
9
1000
=
10
589
1000
=
10589
1000
(5)
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 0,3˙0,3˙ in the form abab (where aa and bb are integers) then we would proceed as follows
x
=
0
,
33333
...
10
x
=
3
,
33333
...
multiply
by
10
on
both
sides
9
x
=
3
(
subtracting
the
second
equation
from
the
first
equation
)
x
=
3
9
=
1
3
x
=
0
,
33333
...
10
x
=
3
,
33333
...
multiply
by
10
on
both
sides
9
x
=
3
(
subtracting
the
second
equation
from
the
first
equation
)
x
=
3
9
=
1
3
(6)
And another example would be to write
5,
4˙
3˙
2˙
5,
4˙
3˙
2˙
as a rational fraction.
x
=
5
,
432432432
...
1000
x
=
5432
,
432432432
...
multiply
by
1000
on
both
sides
999
x
=
5427
(
subtracting
the
second
equation
from
the
first
equation
)
x
=
5427
999
=
201
37
x
=
5
,
432432432
...
1000
x
=
5432
,
432432432
...
multiply
by
1000
on
both
sides
999
x
=
5427
(
subtracting
the
second
equation
from
the
first
equation
)
x
=
5427
999
=
201
37
(7)
For the first example, the decimal was multiplied by 10 and for the second example, the decimal was multiplied by 1000. This is because for the first 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...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.
- Write the following using the repeated decimal notation:
- 0,11111111...0,11111111...
- 0,1212121212...0,1212121212...
- 0,123123123123...0,123123123123...
- 0,11414541454145...0,11414541454145...
- Write the following in decimal form, using the repeated decimal notation:
- 2323
- 13111311
- 456456
- 219219
- Write the following decimals in fractional form:
- 0,6333˙0,6333˙
- 5,313131¯5,313131¯
- 0,999999˙0,999999˙
- Real numbers can be either rational or irrational.
- A rational number is any number which can be written as
a
b
a
b
where aa and bb are integers and b≠0b≠0
- The following are rational numbers:
- Fractions with both denominator and numerator as integers.
- Integers.
- Decimal numbers that end.
- Decimal numbers that repeat.
- If aa is an integer, bb is an integer and cc is irrational, which of the following are rational numbers?
Click here for the
solution
- Write each decimal as a simple fraction:
- 0,50,5
- 0,120,12
- 0,60,6
- 1,591,59
- 12,277˙12,277˙
- Show that the decimal 3,211˙8˙3,211˙8˙ is a rational number.
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solution
- Express 0,78˙0,78˙ as a fraction abab where a,b∈Za,b∈Z (show all working).
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solution
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