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Introduction

As described in 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.

Figure 1
Khan Academy video on Integers and Rational Numbers

The Big Picture of Numbers

Figure 2
Figure 2 (MG10C3_001.png)

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, ....)

Definition

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 b0b0.

Note that because we can write a - b a - b as -a b -a b (in other words, one can always find an equivalent rational expression where b>0b>0) mathematicians typically define rational numbers not as both aa and bb being integers, but rather that aa is an integer and bb is a natural number. This avoids having to worry about zero in the denominator.

Tip:

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)

Rational Numbers

  1. If aa is an integer, bb is an integer and cc is irrational, which of the following are rational numbers?
    1. 5656
    2. a3a3
    3. b2b2
    4. 1c1c
    Click here for the solution
  2. If a1a1 is a rational number, which of the following are valid values for aa?
    1. 1
    2. -10-10
    3. 22
    4. 2,12,1
    Click here for the solution

Forms of Rational Numbers

All integers and fractions with integer numerators and denominators are rational numbers. There are two more forms of rational numbers.

Investigation : Decimal Numbers

You can write the rational number 1212 as the decimal number 0,5. Write the following numbers as decimals:

  1. 1 4 1 4
  2. 1 10 1 10
  3. 2 5 2 5
  4. 1 100 1 100
  5. 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:

  1. decimal numbers that end or terminate, for example the fraction 410410 can be written as 0,4.
  2. 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.

Tip: Notation for Repeating Decimals:

You can use a bar over the repeated numbers to indicate that the decimal is a repeating decimal.

Converting Terminating Decimals into Rational Numbers

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)

Fractions

  1. Write the following as fractions:
    1. 0,10,1
    2. 0,120,12
    3. 0,580,58
    4. 0,25890,2589
    Click here for the solution

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 0,3˙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.

Repeated Decimal Notation

  1. Write the following using the repeated decimal notation:
    1. 0,11111111...0,11111111...
    2. 0,1212121212...0,1212121212...
    3. 0,123123123123...0,123123123123...
    4. 0,11414541454145...0,11414541454145...
    Click here for the solution
  2. Write the following in decimal form, using the repeated decimal notation:
    1. 2323
    2. 13111311
    3. 456456
    4. 219219
    Click here for the solution
  3. Write the following decimals in fractional form:
    1. 0,6333˙0,6333˙
    2. 5,313131¯5,313131¯
    3. 0,999999˙0,999999˙
    Click here for the solution

Summary

  • 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 b0b0
  • The following are rational numbers:
    1. Fractions with both denominator and numerator as integers.
    2. Integers.
    3. Decimal numbers that end.
    4. Decimal numbers that repeat.

End of Chapter Exercises

  1. If aa is an integer, bb is an integer and cc is irrational, which of the following are rational numbers?
    1. 5656
    2. a3a3
    3. b2b2
    4. 1c1c
    Click here for the solution
  2. Write each decimal as a simple fraction:
    1. 0,50,5
    2. 0,120,12
    3. 0,60,6
    4. 1,591,59
    5. 12,277˙12,277˙
    Click here for the solution
  3. Show that the decimal 3,211˙8˙3,211˙8˙ is a rational number.
    Click here for the solution
  4. Express 0,78˙0,78˙ as a fraction abab where a,bZa,bZ (show all working).
    Click here for the solution

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