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Basic 9 Β· Mathematics Β· Number and Numeration System

The Rational Number System and Sets

In this topic you will solve problems on the relationship between members of the rational number system and apply the concept of sets to these numbers and to real-life problems.

By the end of these notes you should be able to:

  • Describe the relationships within the rational number system
  • Convert between fractions and decimals
  • Apply sets to number problems

What you already know

  • Sets and types of numbers

Copy everything that has a blue line on its left into your notebook. The other parts are for reading.

Lesson 1

Strand
Number
Sub-strand
Number and Numeration System
Indicator
B9.1.1.2.1 – Solve problems on relationship between members of the rational number system using knowledge and understanding of the concept of union and intersection of two sets

Introduction

Counting numbers came first. As people needed to deal with debts and shares, they invented new kinds of number.

1. Sets of Numbers

The number system
SetSymbolExamples
natural numbersβ„•1, 2, 3, ...
whole numbersπ•Ž0, 1, 2, 3, ...
integersβ„€..., βˆ’2, βˆ’1, 0, 1, 2, ...
rational numbersβ„šΒ½, βˆ’3, 0.25, 0.333...
irrational numbers√2, Ο€

Every natural number is a whole number, every whole number is an integer and every integer is rational. So each set sits inside the next: β„• βŠ‚ π•Ž βŠ‚ β„€ βŠ‚ β„š. We use the symbol βŠ‚ to mean "is a subset of".

2. What Is a Rational Number?

Rational number: A number that can be written as a/b, where a and b are integers and b is not zero.

5 is rational, as it equals 5/1. βˆ’3 equals βˆ’3/1. 0.75 equals ΒΎ. Numbers such as √2 and Ο€ are irrational. Their decimals never end, and never repeat. Together, rational and irrational numbers form the real numbers.

Classify

Classify 7, βˆ’4, β…” and √3.

  1. 7 is natural, whole, integer and rational.
  2. βˆ’4 is an integer and rational.
  3. β…” is rational.
  4. √3 is irrational.

Answer: 7 belongs to all four sets, βˆ’4 to two, β…” to one, and √3 is irrational

A number line on a classroom wall
Figure 1: Number line Draw a diagram of the number sets.

3. Numbers on a Number Line

All rational numbers can be placed on a number line. Between any two rationals, there is always another. For example, between Β½ and ΒΎ lies ⅝, which is their mean. Negative numbers lie to the left of zero. Numbers increase as you move to the right, so βˆ’2 is greater than βˆ’5. This often confuses learners, but it makes sense if you think of temperatures or debts.

4. Why These Numbers Matter

Each kind of number solves a problem. Natural numbers count. Zero shows nothing. Integers show debts, depths and temperatures below zero. Rationals show shares and measurements. Irrationals appear in circles and in the diagonals of squares. Understanding how they are linked will help you with algebra and geometry.

New words

rational number
A number of the form a/b.
integer
A whole number or its negative.
irrational number
A number that is not rational.
subset
A set inside another set.

Summary

  • Natural numbers, whole numbers and integers are all rational.
  • A rational number can be written as a/b.
  • √2 and Ο€ are irrational.

Class activity: Number Sort

  1. Write numbers on cards.
  2. Sort them into a nested diagram.
  3. Explain each placement.

You need: Bottle tops, Two raffia hoops, Number cards

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. Which is an integer?

    • A. βˆ’4
    • B. Β½
    • C. 0.5
    • D. Ο€
    Show answer
    Answer: βˆ’4Whole.

  2. 2. Which is irrational?

    • A. √2
    • B. ΒΎ
    • C. 0.25
    • D. 7
    Show answer
    Answer: √2Non-repeating.

  3. 3. Every integer is rational.

    • A. True
    • B. False
    Show answer
    Answer: Truea/1.

  4. 4. β„€ stands for the set of ______.

    Show answer
    Answer: integersSymbol.

  5. 5. Find a number between β…“ and Β½.

    Show answer
    Answer: 5/12 (their mean)Mean.

  6. 6. Explain why 0 is rational.

    Show answer
    Answer: It can be written as 0/1.Definition.

Lesson 2

Strand
Number
Sub-strand
Number and Numeration System
Indicator
B9.1.1.2.1 – Solve problems on relationship between members of the rational number system using knowledge and understanding of the concept of union and intersection of two sets

Introduction

ΒΌ and 0.25 are the same number in different dress. Moving between them is a key skill.

1. Fractions to Decimals

To change a fraction into a decimal, divide the numerator by the denominator. Some decimals terminate, or stop: ΒΎ = 0.75. Others recur, or repeat forever: β…“ = 0.333..., which we write with a dot over the 3. A fraction in its lowest terms gives a terminating decimal only if the prime factors of its denominator are 2s and 5s.

Fractions and decimals
FractionDecimalType
Β½0.5terminating
β…›0.125terminating
β…“0.333...recurring
1/70.142857142857...recurring
2/110.181818...recurring

2. Terminating Decimals to Fractions

Converting

Write 0.375 as a fraction in its lowest terms.

  1. 0.375 = 375/1000.
  2. Divide the top and the bottom by 125.

Answer: 3/8

3. Recurring Decimals to Fractions

One recurring digit

Write 0.444... as a fraction.

  1. Let x = 0.444...
  2. Then 10x = 4.444...
  3. Subtract: 9x = 4.

Answer: x = 4/9

Two recurring digits

Write 0.121212... as a fraction.

  1. Let x = 0.1212...
  2. Then 100x = 12.1212...
  3. Subtract: 99x = 12.

Answer: x = 12/99 = 4/33

Fraction circles cut into equal parts
Figure 2: Fraction circles Convert five fractions into decimals.

4. Comparing Rational Numbers

To compare fractions and decimals, change all of them to one form. Put ⅝, 0.6 and 65% in order: as decimals they are 0.625, 0.6 and 0.65, so the order from smallest is 0.6, ⅝, 65%. Decimals are usually the simplest form for comparing. Use two or three decimal places, and fill in end zeros to help you to compare.

New words

terminating decimal
A decimal that stops.
recurring decimal
A decimal that repeats forever.
numerator
The top number of a fraction.
denominator
The bottom number of a fraction.

Summary

  • Divide to change a fraction to a decimal.
  • Rational numbers terminate or recur.
  • Algebra converts recurring decimals into fractions.

Class activity: Decimal Patterns

  1. Use a calculator to find 1/7, 2/7 and 3/7.
  2. Look for patterns in the digits.
  3. Share what you find.

You need: Bottle tops, Two raffia hoops, Number cards

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. ΒΎ as a decimal is…

    • A. 0.75
    • B. 0.34
    • C. 0.43
    • D. 1.33
    Show answer
    Answer: 0.753 Γ· 4.

  2. 2. Which is recurring?

    • A. β…“
    • B. ΒΌ
    • C. Β½
    • D. β…›
    Show answer
    Answer: β…“0.333...

  3. 3. Rational numbers can have recurring decimals.

    • A. True
    • B. False
    Show answer
    Answer: Trueβ…“.

  4. 4. 0.5 as a fraction is ______.

    Show answer
    Answer: Β½Convert.

  5. 5. Write 0.777... as a fraction.

    Show answer
    Answer: 7/99x = 7.

  6. 6. Arrange in ascending order: 0.7, β…”, 72%.

    Show answer
    Answer: β…”, 0.7, 72%0.667, 0.7, 0.72.

Lesson 3

Strand
Number
Sub-strand
Number and Numeration System
Indicator
B9.1.1.2.2 – Apply the concept of sets to solve problems on relationship between members of rational number system and solve real life problems involving union and intersection of two sets

Introduction

Sets help us to organise numbers and people. Venn diagrams make relationships easy to see.

1. Review of Sets

Set notation
SymbolMeaningExample
∈is a member of3 ∈ {1, 2, 3}
βˆͺunion{1, 2} βˆͺ {2, 3} = {1, 2, 3}
∩intersection{1, 2} ∩ {2, 3} = {2}
n(A)number of membersn({a, b}) = 2
Aβ€²complementmembers not in A
βˆ…empty setno members

2. Sets of Numbers

Number sets

U = {1, 2, 3, ..., 12}. A = {even numbers}, B = {multiples of 3}. Find A ∩ B and n(A βˆͺ B).

  1. A = {2, 4, 6, 8, 10, 12}, B = {3, 6, 9, 12}.
  2. A ∩ B = {6, 12}.
  3. n(A βˆͺ B) = 6 + 4 βˆ’ 2.

Answer: {6, 12} and 8

3. Word Problems

Survey

In a class of 40, 25 play football, 18 play volleyball and 7 play neither. How many play both?

  1. Those who play at least one: 40 βˆ’ 7 = 33.
  2. n(F βˆͺ V) = n(F) + n(V) βˆ’ n(F ∩ V).
  3. 33 = 25 + 18 βˆ’ x.
  4. x = 43 βˆ’ 33.

Answer: 10 learners

A Venn diagram shows this clearly. Put the 10 in the middle first, then 15 for football only, 8 for volleyball only and 7 outside the circles. Check: 10 + 15 + 8 + 7 = 40.

Students playing sport at school
Figure 3: School sports Survey your class and draw a Venn diagram.

4. Rational Numbers in Sets

We can describe sets of rationals using rules. For example, {x : 0 < x < 1} contains all numbers between 0 and 1, and this set is infinite. Probability uses sets too. If a learner is chosen at random from the class above, the probability of choosing someone who plays both sports is 10/40 = ΒΌ. Such links show that sets, fractions and data all belong to one mathematical family.

New words

union
All members in either set.
intersection
Members in both sets.
complement
Members not in a set.
universal set
The set of everything considered.

Summary

  • Sets organise numbers and people.
  • n(A βˆͺ B) = n(A) + n(B) βˆ’ n(A ∩ B).
  • Start Venn diagrams from the intersection.

Class activity: Class Venn

  1. Survey learners about two subjects.
  2. Draw a Venn diagram.
  3. Calculate the intersection.

You need: Bottle tops, Two raffia hoops, Number cards

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. A ∩ B means…

    • A. members in both sets
    • B. all members
    • C. no members
    • D. complement
    Show answer
    Answer: members in both setsNotation.

  2. 2. n(A) = 8, n(B) = 6, n(A ∩ B) = 3. n(A βˆͺ B) =

    • A. 11
    • B. 17
    • C. 14
    • D. 5
    Show answer
    Answer: 118 + 6 βˆ’ 3.

  3. 3. βˆ… means the empty set.

    • A. True
    • B. False
    Show answer
    Answer: TrueSymbol.

  4. 4. Members in either set form the ______.

    Show answer
    Answer: unionTerm.

  5. 5. U = {1 to 10}, A = {odd numbers}. List Aβ€².

    Show answer
    Answer: {2, 4, 6, 8, 10}Complement.

  6. 6. Of 50 learners, 30 like Maths, 25 like Science and 10 like both. How many like neither?

    Show answer
    Answer: 550 βˆ’ 45.

Homework

  1. Create and solve two Venn diagram word problems.
The Rational Number System and Sets – Basic 9 Mathematics Notes | sirbus