Unit outcomes

By the end of this unit you will be able to:

• Identify rational numbers and convert between terminating and recurring decimals such as [latex]\scriptsize \displaystyle \frac{a}{b};a,b\in \mathbb{Z},~b\ne 0[/latex].
• Round off rational and irrational numbers to an appropriate degree of accuracy.

What you should know

Before you start this unit, make sure you can:

• Classify real numbers as natural numbers, whole numbers, or integers.
• Perform calculations using order of operations on fractions.

Test your understanding of real numbers and operations by trying these questions before continuing with this unit.

1. Classify each number below as a natural number ([latex]\scriptsize \mathbb{N}[/latex]), whole number ([latex]\scriptsize {{\mathbb{N}}_{0}}[/latex]) or integer ([latex]\scriptsize \mathbb{Z}[/latex]). Note: List all possible classifications for each number.
   1. [latex]\scriptsize \sqrt{9}[/latex]
   2. [latex]\scriptsize -2[/latex]
   3. [latex]\scriptsize \displaystyle \frac{125}{25}[/latex]
2. Evaluate:
   1. [latex]\scriptsize 24+36(\displaystyle \frac{2}{3})[/latex]
   2. [latex]\scriptsize (3\cdot 2)(3\cdot 2)-4(6+2)[/latex]

Solutions

1. 1. [latex]\scriptsize \sqrt{9}=3[/latex] is [latex]\scriptsize \mathbb{N}[/latex], [latex]\scriptsize {{\mathbb{N}}_{0}}[/latex] and [latex]\scriptsize \mathbb{Z}[/latex].
   2. [latex]\scriptsize -2[/latex] is an [latex]\scriptsize \mathbb{Z}[/latex]
   3. [latex]\scriptsize \displaystyle \frac{125}{25}=5[/latex] is [latex]\scriptsize \mathbb{N}[/latex], [latex]\scriptsize {{\mathbb{N}}_{0}}[/latex] and [latex]\scriptsize \mathbb{Z}[/latex]
2. 1. .
      [latex]\scriptsize \begin{align} 24+36\left( \displaystyle \frac{2}{3} \right)& =24+12(2) \\ & =48 \end{align}[/latex]
   2. .
      [latex]\scriptsize \begin{align} (3\cdot 2)(3\cdot 2)-4(6+2)& =(6)(6)-4(8) \\ & =36-32 \\ & =4 \end{align}[/latex]

Introduction

The earliest use of numbers was to count items. Farmers, cattlemen and tradesmen used tokens, stones or markers to signify a single quantity, for example a sheaf of grain, a head of livestock or a fixed length of cloth. Doing so made commerce possible, leading to improved communication and the spread of civilization.

About four thousand years ago, Egyptians introduced fractions to the number system. They first used them to show reciprocals. The reciprocal of a number is [latex]\scriptsize 1[/latex] divided by the number. Later, they used them to represent the amount when a quantity was divided into equal parts.

In this unit, we will explore the number system further by using rational and irrational numbers.

The number system revised

We will begin with a recap of the number system.

Complex numbers include all real and imaginary (non-real) numbers. More about complex numbers is covered in Level 3. For now we will focus on real numbers, which are made up of rational and irrational numbers.

Beginning with the natural numbers, we expand each set to form a larger set of numbers.

We use the following definitions:

The set of natural numbers [latex]\scriptsize \mathbb{N}[/latex] includes the numbers used for counting: [latex]\scriptsize \left\{ 1,\text{ }2,\text{ }3,\text{ }... \right\}[/latex].

The set of whole numbers [latex]\scriptsize {{\mathbb{N}}_{0}}[/latex] is the set of natural numbers plus zero: [latex]\scriptsize \left\{ 0,\text{ }1,\text{ }2,\text{ }3,\text{ }... \right\}[/latex].

The set of integers [latex]\scriptsize \mathbb{Z}[/latex] adds the negative of the natural numbers to the set of whole numbers [latex]\scriptsize \left\{ ...,\text{ }-3,\text{ }-2,\text{ }-1,\text{ }0,\text{ }1,\text{ }2,\text{ }3,\text{ }...\text{ } \right\}[/latex]. It is important to remember that integral values are not fractions.

The set of rational numbers [latex]\scriptsize \mathbb{Q}[/latex] includes fractions that can be written as [latex]\scriptsize \displaystyle \frac{a}{b}[/latex] where [latex]\scriptsize a,b\in \mathbb{Z},~b\ne 0[/latex].

The set of irrational numbers [latex]\scriptsize {\mathbb{Q}}'[/latex] are numbers that cannot be written as a fraction with the numerator and denominator as integers. These fractions give us nonrepeating and nonterminating (they do not end) decimals.

Rational and irrational numbers

We can see from the definition that every natural number, whole number and integer is a rational number with a denominator of [latex]\scriptsize 1[/latex].

As they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

a terminating decimal ([latex]\scriptsize \displaystyle \frac{3}{4}=0.75[/latex]) or

a repeating decimal ([latex]\scriptsize \displaystyle \frac{4}{11}=\text{0}\text{.36363636}...=0.\overline{36}[/latex]).

We use a line drawn over the repeating block of numbers to show that those numbers are repeated.

The following numbers are examples of rational numbers:

[latex]\scriptsize \displaystyle \frac{5}{1},\displaystyle \frac{15}{3},\displaystyle \frac{-1}{3},\displaystyle \frac{2}{3},\displaystyle \frac{12}{24}[/latex]

We can see that all the numerators and denominators are integers. You can write any rational number as a decimal number but not all decimal numbers are rational numbers.

Exercise 1.1

1. Write each of the following as a rational number:
   1. [latex]\scriptsize 6[/latex]
   2. [latex]\scriptsize -5[/latex]
   3. [latex]\scriptsize 0[/latex]
2. Use your calculator to determine if the following rational numbers are terminating or recurring decimals.
   1. [latex]\scriptsize \displaystyle \frac{-5}{6}[/latex]
   2. [latex]\scriptsize \displaystyle \frac{325}{13}[/latex]
   3. [latex]\scriptsize \displaystyle \frac{13}{25}[/latex]

The full solutions are at the end of the unit.

The following numbers are examples of irrational numbers.

[latex]\scriptsize \sqrt{2},\sqrt{3},\pi ,\displaystyle \frac{1}{\sqrt{2}}[/latex]

These are not rational numbers, because either the numerator or the denominator is not an integer.

To identify whether a number is rational or irrational, first write the number in decimal form.

If the number terminates then it is rational. If it goes on forever, then look for a repeated pattern of digits. If there is no repeated pattern, then the number is irrational.

Exercise 1.2

1. Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.
   1. [latex]\scriptsize \sqrt{9}[/latex]
   2. [latex]\scriptsize \sqrt{10}[/latex]
   3. [latex]\scriptsize \displaystyle \frac{17}{34}[/latex]
   4. [latex]\scriptsize \text{0}\text{.3033033303333}[/latex]

The full solutions are at the end of the unit.

Convert terminating decimals to rational numbers

Any terminating decimal can be written as a decimal divided by one. For example, [latex]\scriptsize 0.45=\displaystyle \frac{0.45}{1}[/latex].

To convert [latex]\scriptsize \displaystyle \frac{0.45}{1}[/latex]to a rational number, multiply both the numerator and denominator by [latex]\scriptsize 100[/latex] (because there are two digits after the decimal point so that is [latex]\scriptsize 10\times 10=100[/latex]).

[latex]\scriptsize \displaystyle \frac{0.45}{1}\times \displaystyle \frac{100}{100}=\displaystyle \frac{45}{100}[/latex]

Do you see that the numerator is now a whole number? If you multiply the numerator and denominator by the correct power of ten, you will always end up with a whole number in the numerator.

We can simplify further by dividing both the numerator and denominator by five, to get:

[latex]\scriptsize \displaystyle \frac{45}{100}=\displaystyle \frac{9}{20}[/latex]

Take note!

These are the steps you can use to convert a terminating decimal to a rational number.

1. Rewrite the terminating decimal as a decimal divided by one.
2. Multiply both the numerator and denominator by [latex]\scriptsize 10[/latex] for every number after the decimal point in the numerator. For example, if there are three numbers after the decimal point, then multiply by [latex]\scriptsize 1\text{ }000[/latex] , if there are four then multiply by, [latex]\scriptsize 10\text{ 000}[/latex] and so on.
3. Simplify the fraction if possible.

When there is a whole number before the decimal point, set the whole number aside and bring it back in at the end. As shown in the next example.

Example 1.1

Write [latex]\scriptsize 10.585[/latex] as a mixed fraction.

Set the whole number [latex]\scriptsize 10[/latex] aside and remember to bring it back in at the end.

There are three zeroes after the decimal so multiply both the numerator and denominator by [latex]\scriptsize 1~000[/latex].

[latex]\scriptsize \begin{align} \displaystyle \frac{0.585}{1}& =\displaystyle \frac{0.585\times 1~000}{1~000} \\ & =\displaystyle \frac{585}{1~000} \end{align}[/latex]

Simplify the fraction if possible.

[latex]\scriptsize \displaystyle \frac{585\div 5}{1~000\div 5}=\displaystyle \frac{117}{200}[/latex]

Bring the [latex]\scriptsize 10[/latex] back in and write the answer as a mixed fraction.

[latex]\scriptsize 10.585=10\displaystyle \frac{117}{200}[/latex]

Note: Do not ‘add’ the [latex]\scriptsize 10[/latex] to the fraction, you will get an incorrect answer. Simply put the whole number back in front of the fraction.

Exercise 1.3

1. Convert the following decimals to fractions:
   1. [latex]\scriptsize 0.2589[/latex]
   2. [latex]\scriptsize 5.24[/latex]

The full solutions are at the end of the unit.

Activity 1.1: Convert recurring decimal numbers into fractions

Time required: 15 minutes

What you need:

• pen and paper

What to do:

Write [latex]\scriptsize 0.\dot{3}[/latex] in the form [latex]\scriptsize \displaystyle \frac{a}{b}[/latex] where [latex]\scriptsize a[/latex] and [latex]\scriptsize b[/latex] are integers.

1. We have [latex]\scriptsize 0.\dot{3}=0.3333...[/latex] a repeating decimal. Let [latex]\scriptsize x=0.3333...[/latex]. We will call this equation one.
2. Multiply both sides of equation one by [latex]\scriptsize 10[/latex]. We will call this equation two.
3. Subtract equation one from equation two.
4. Simplify and solve for [latex]\scriptsize x[/latex].
5. Write [latex]\scriptsize 0.\dot{3}[/latex] in the form [latex]\scriptsize \displaystyle \frac{a}{b}[/latex].

What did you find?

1. 1. Equation one: [latex]\scriptsize x=0.3333...[/latex]
   2. The decimal was multiplied by [latex]\scriptsize 10[/latex] because there was only one digit ([latex]\scriptsize 3[/latex]) recurring after the decimal point. In general, if you have one digit recurring, then multiply by [latex]\scriptsize 10[/latex]. If you have two digits recurring, then multiply by [latex]\scriptsize 100[/latex]. If you have three digits recurring, then multiply by [latex]\scriptsize 1~000[/latex], and so on.
      You should have the following for equation two: [latex]\scriptsize 10x=3.3333...[/latex]
   3. We want the repeating parts of the decimal to cancel out when subtracting the equations. By subtracting equation one from equation two you will get:

[latex]\scriptsize \begin{align} 10x& =3.3333... \\ -x& =-0.3333... \\ \therefore 9x& =3 \end{align}[/latex]

4. 4. You can simplify by dividing both sides of the equation by nine.

[latex]\scriptsize \begin{align} \therefore x& = \displaystyle \frac{3}{9} \\ & =\displaystyle \frac{1}{3} \end{align}[/latex]

1. 5. Since we started out by defining [latex]\scriptsize x=0.3333...[/latex] this means that [latex]\scriptsize 0.3333...=\displaystyle \frac{1}{3}[/latex].

Exercise 1.4

1. Convert the following decimals to fractions:
   1. [latex]\scriptsize 0.\overline{25}[/latex]
   2. [latex]\scriptsize 0.8\dot{3}[/latex]

The full solutions are at the end of the unit.

Rounding off decimals

Rounding off a decimal number to a given number of decimal places is the quickest way to approximate a number.

For example, if you wanted to round off [latex]\scriptsize 2.6525272[/latex] to three decimal places, you would:

count three places after the decimal place and assess the value of the fourth digit

round up the third digit if the fourth digit is greater than or equal to [latex]\scriptsize 5[/latex]

leave the third digit unchanged if the fourth digit is less than [latex]\scriptsize 5[/latex]

if the third digit is [latex]\scriptsize 9[/latex] and needs to be rounded up, then the [latex]\scriptsize 9[/latex] becomes a [latex]\scriptsize 0[/latex] and the second digit is rounded up.

So, since the fourth digit after the decimal point in [latex]\scriptsize 2.6525272[/latex] is a [latex]\scriptsize 5[/latex] , we must round up the digit in the third decimal place to [latex]\scriptsize 3[/latex] . So the final answer of [latex]\scriptsize 2.6525272[/latex] rounded to three decimal places is [latex]\scriptsize 2.653[/latex] .

Exercise 1.5

1. Round off the following numbers to the indicated number of decimal places:
   1. [latex]\scriptsize \displaystyle \frac{131}{9}[/latex] to three decimal places
   2. [latex]\scriptsize \pi[/latex] to six decimal places
   3. [latex]\scriptsize \sqrt{2}[/latex] to four decimal places.

The full solutions are at the end of the unit.

Summary

In this unit you have learnt the following:

• To identify and differentiate between rational and irrational numbers.
• Convert terminating decimals to rational numbers.
• Convert recurring decimals to rational numbers.
• How to round off to a certain number of decimal places to approximate a number.

Unit 1: Assessment

Suggested time to complete: 30 minutes

1. Which of the statements is true?
   1. Every integer is a natural number.
   2. Every natural number is a whole number.
   3. There are no decimals in whole numbers.
2. State whether the following numbers are rational or irrational. If the number is rational, state whether it is a natural number, whole number or an integer.
   1. [latex]\scriptsize \displaystyle \frac{5}{8}[/latex]
   2. [latex]\scriptsize \text{0,651268962154862}...[/latex]
   3. [latex]\scriptsize \displaystyle \frac{\sqrt{16}}{4}[/latex]
   4. [latex]\scriptsize \sqrt[3]{4}[/latex]
3. Write the following as fractions:
   1. [latex]\scriptsize 0.68[/latex]
   2. [latex]\scriptsize 2.\dot{3}[/latex]
   3. [latex]\scriptsize 0.\overline{12}[/latex]
   4. [latex]\scriptsize 0.6\dot{3}[/latex]
4. Write the following in decimal form, using the recurring decimal notation and without using a calculator:
   1. [latex]\scriptsize \displaystyle \frac{7}{33}[/latex]
   2. [latex]\scriptsize 1\displaystyle \frac{3}{11}[/latex]
   3. [latex]\scriptsize 2\displaystyle \frac{1}{9}[/latex]
5. Round off the following to 3 decimal places:
   1. [latex]\scriptsize \text{12,56637061}...[/latex]
   2. [latex]\scriptsize \text{0,05555555}...[/latex]
   3. [latex]\scriptsize \text{0,2666666}...[/latex]
6. Study the diagram below
   1. Calculate the area of ABDE to two decimal places.
   2. Calculate the area of BCD to two decimal places.
   3. Using your answers in a) and b), calculate the area of ABCDE.

The full solutions are at the end of the unit.

Unit 1: Solutions

Exercise 1.1

1. Write a fraction with the integer in the numerator and [latex]\scriptsize 1[/latex] in the denominator.
   1. [latex]\scriptsize 6=\displaystyle \frac{6}{1}[/latex]
   2. [latex]\scriptsize -5=\displaystyle \frac{-5}{1}[/latex]
   3. [latex]\scriptsize 0=\displaystyle \frac{0}{1}[/latex]
2. Write each fraction as a decimal by dividing the numerator by the denominator.
   1. [latex]\scriptsize \displaystyle \frac{-5}{6}=-\text{0}\text{.8}\overline{3}[/latex] repeating decimal
   2. [latex]\scriptsize \displaystyle \frac{325}{13}=25[/latex] (or [latex]\scriptsize 25.0[/latex]) a terminating decimal
   3. [latex]\scriptsize \displaystyle \frac{13}{25}=0.52[/latex] a terminating decimal

Back to Exercise 1.1

Exercise 1.2

1. 1. [latex]\scriptsize \sqrt{9}=3[/latex] is a terminating rational number
   2. [latex]\scriptsize \sqrt{10}=3.16227766016837933199...[/latex] The average scientific calculator can only calculate up to ten decimal places. Using a powerful scientific calculator we can get a more precise answer. [latex]\scriptsize \therefore \sqrt{10}[/latex] is irrational.
   3. [latex]\scriptsize \displaystyle \frac{17}{34}=\displaystyle \frac{1}{2}=0.5[/latex] is a terminating rational number
   4. [latex]\scriptsize \text{0}\text{.3033033303333}...[/latex] is not a terminating decimal. Also note that there is no repeating pattern because the group of threes increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.

Back to Exercise 1.2

Exercise 1.3

1. 1. .
      [latex]\scriptsize \begin{align} \displaystyle \frac{0.2589}{1}& =\displaystyle \frac{0.2589}{1}\times \displaystyle \frac{10\text{ 000}}{10\text{ 000}} \\ & =\displaystyle \frac{2\text{ }589}{10\text{ 000}} \end{align}[/latex]
   2. Work with only the decimal and bring the whole number back in at the end.
      [latex]\scriptsize \begin{align} \displaystyle \frac{0.24}{1}& =\displaystyle \frac{0.24\times 100}{100} \\ & =\displaystyle \frac{24}{100} \\ & =\displaystyle \frac{6}{25}\\ \therefore 5.24& =5\displaystyle \frac{6}{25} \end{align}[/latex]

Back to Exercise 1.3

Exercise 1.4

1. 1. .
      [latex]\scriptsize \begin{align*} \text{Let }x&=0.2525&&\text{Equation 1}\\ 100x&=25.2525&&\text{Equation 2}\\ 99x&=25&&\text{Subtract equation 1 from equation 2}\\ x&=\displaystyle\frac{25}{99}\\ \therefore 0.\overline{25}&=\displaystyle\frac{25}{99} \end{align*}[/latex]
   2. .
      [latex]\scriptsize \begin{align*} \text{Let }x&=8.83333\\ 10x&=8.3333&&\text{Equation 1}\\ 100x&=83.333&&\text{Equation 2}\\ 90x&=75&&\text{Subtract equation 1 from equation 2}\\ x&=\displaystyle\frac{75}{90}\\ &=\displaystyle\frac{5}{6}\\ \therefore 0.8\dot{3}&=\displaystyle\frac{5}{6} \end{align*}[/latex]

Back to Exercise 1.4

Exercise 1.5

1. 1. [latex]\scriptsize \displaystyle \frac{131}{9}=14.556[/latex] to three decimal places.
   2. [latex]\scriptsize \pi =3.141593[/latex] to six decimal places.
   3. [latex]\scriptsize \sqrt{2}=1.4142[/latex] to four decimal places.

Back to Exercise 1.5

Unit 1: Assessment

1. Statements b) and c) are true.
2. 1. [latex]\scriptsize \displaystyle \frac{5}{8}[/latex] is a rational number [latex]\scriptsize (\mathbb{Q})[/latex] but not also an integer [latex]\scriptsize (\mathbb{Z})[/latex].
   2. [latex]\scriptsize \text{0,651268962154862}...[/latex] is a non-repeating non-terminating decimal. Therefore it is irrational number [latex]\scriptsize ({\mathbb{Q}}')[/latex]
   3. [latex]\scriptsize \displaystyle \frac{\sqrt{16}}{4}=1[/latex] therefore it is a rational number [latex]\scriptsize (\mathbb{Q})[/latex]. But it is also an integer [latex]\scriptsize (\mathbb{Z})[/latex], a whole number [latex]\scriptsize ({{\mathbb{N}}_{0}})[/latex] and a natural number [latex]\scriptsize (\mathbb{N})[/latex].
   4. [latex]\scriptsize \sqrt[3]{4}[/latex] is an irrational number [latex]\scriptsize ({\mathbb{Q}}')[/latex].
3. 1. .
      [latex]\scriptsize \begin{align} 0.68& =\displaystyle \frac{0.68}{1}\times \displaystyle \frac{100}{100} \\ & =\displaystyle \frac{68}{100} \\ & =\displaystyle \frac{17}{25} \end{align}[/latex]
   2. .
      [latex]\scriptsize \begin{align} \text{Let }x& =2.3333 \\ 10x& =23.333 \\ \therefore 9x& =23.333-2.333 \\ 9x& =21 \\ \therefore x& =\displaystyle \frac{21}{9}=\displaystyle \frac{7}{3} \\ \therefore 2.\dot{3}& =\displaystyle \frac{7}{3} \end{align}[/latex]
   3. .
      [latex]\scriptsize 0.\overline{12}=0.121212[/latex]
      [latex]\scriptsize \begin{align} \text{Let }x& =0.1212 \\ \therefore 100x& =12.1212 \\ \therefore 99x& =(12.1212-0.1212) \\ \therefore 99x& =12 \\ \therefore x& =\displaystyle \frac{12}{99} \\ & =\displaystyle \frac{4}{33} \\ \therefore 0.\overline{12}& =\displaystyle \frac{4}{33} \end{align}[/latex].
   4. .
      [latex]\scriptsize 0.6\dot{3}=0.6333[/latex]
      [latex]\scriptsize \begin{align} \text{Let }x& =0.6333 \\ 10x& =6.333 \\ 100x& =63.333 \\ \therefore 90x& =(63.333-6.333) \\ & =57 \\ \therefore x& =\displaystyle \frac{57}{90} \\ & =\displaystyle \frac{19}{30} \\ \therefore 0.63& =\displaystyle \frac{19}{30} \end{align}[/latex]
4. 1. [latex]\scriptsize \displaystyle \frac{7}{33}=0.\overline{21}[/latex]
   2. .
      [latex]\scriptsize \begin{align} 1\displaystyle \frac{3}{11}& =\displaystyle \frac{14}{11} \\ & =1.\overline{27}3 \end{align}[/latex]
   3. .
      [latex]\scriptsize \begin{align} 2\displaystyle \frac{1}{9}& =\displaystyle \frac{19}{9} \\ & =2.\dot{1} \end{align}[/latex]
5. 1. [latex]\scriptsize 12.56637061...=12.566[/latex] correct to [latex]\scriptsize 3[/latex] decimal places
   2. [latex]\scriptsize \text{0.05555555}...=0.056[/latex] correct to [latex]\scriptsize 3[/latex] decimal places
   3. [latex]\scriptsize \text{0.2666666}...0.267[/latex] correct to [latex]\scriptsize 3[/latex] decimal places
6. 1. Area of ABDE:
      [latex]\scriptsize \pi \times \pi =9.86960...=9.87[/latex] correct to [latex]\scriptsize 2[/latex] decimal places
   2. Area of BCD:
      [latex]\scriptsize \begin{align} \displaystyle \frac{1}{2}b\cdot h& =\displaystyle \frac{1}{2}(\pi )(\pi ) \\ & =4.934802...=4.93\text{ correct to }2\text{ decimal places} \end{align}[/latex]
   3. Area of ABCDE = Area of ABDE + Area of BCD
      [latex]\scriptsize \begin{align} \text{Area }& =9.87+4.93\text{ } \\ & =14.8 \end{align}[/latex]

Back to Unit 1: Assessment