Topic 2: Numbers - Mathematics Form One

Topic 2: Numbers – Mathematics Form One

ABOUT THE TOPIC

Numbers play a vital role in everyday life. Different activities are performed with the help of numbers. Numbers are used to quantify and measure quantities. For instance, length, mass, time, volume, and population are some of the quantities represented by numbers. In this lesson, you will learn about rational, irrational, and real numbers.

You will also learn about repeating decimals, inequalities and absolute values of real numbers. The competencies developed will enable you to perform daily life activities such as counting things, managing money, distributing items, and interpreting numbers based on contexts, and many other applications.

Concept of numbers

Numbers are classified into different categories. Some categories of numbers that you have already learned include whole numbers, natural numbers, fractions, integers, and decimals.

Other major categories of numbers include rational, irrational, and real numbers. Activity 2.1 enables you to identify categories of numbers from daily life activities.

Activity 1: Categorising numbers

1. Use different measuring tools such as tape measure, ruler, weighing balance, and measuring cylinders to measure lengths, masses, and volumes of different objects, respectively.

2. Record the measurements in task 1 and categorise them based on your understanding of categories of numbers.

Rational numbers

Representation of rational numbers on a number line

Rational numbers can be represented on a number line. Positive rational numbers are represented on the right of zero (the origin) and the negative rational numbers on the left of the origin.

The number line helps us to determine other rational numbers between any two rational numbers by increasing the number of divisions. Activity 2.2 enables you to locate numbers on a number line.

Activity 2: Locating numbers on a number line

1. Prepare a number line (2 to 3 metres long) using sticks and masking tapes or any other materials of your choice.

2. Cut manila cards in small sizes and write on them different types of numbers such as whole, integers, and fractions.

3. Give the cards to others to post each of the numbers on appropriate position on the number line.

4. While sticking the cards, explain to others why you think the positions are appropriate.

EXERCISE 1

Repeating decimals

Decimal numbers are part of rational numbers and are common in our daily life activities. The quantities such as length, height, age, volume and mass can be presented in decimals. Activity 2.3 guides you in expressing various quantities in fractions into decimals.

Activity 3: Expressing measurements of quantities in decimals

1. Take some fruits or similar objects and divide them into two, three, four, five, six, and seven equal parts.

2. Convert each fraction in task 1 into decimals in many decimal places as possible. You can work manually or use a calculator.

3. Study carefully the decimal part of the fractions and write down their unique characteristics.

A repeating decimal, also known as recurring decimal is a decimal number with at least one digit in the decimal part that repeats consecutively in a regular order without an end. For example, 1.6666666… and 0.639639639639… are repeating decimals because 6 and 639 digits from the two decimal numbers, respectively, in the decimal part repeat themselves without an end.

The three dots indicate that the repeating digits continue infinitely. Repeating decimals can also be represented by using a dot or a bar that is placed on top of a repeating digit.

Example

In Example above, the digits with a dot or a bar are repeating infinitely. In (b) and (c), it can be observed that if a group of digits is repeating, a dot should be put over the first and the last repeating digits.

Decimals are either terminating or non-terminating. Terminating decimals have a definite number of digits after the decimal point while non-terminating decimals have an endless number of digits after the decimal point.

Thus, repeating decimals are non-terminating with one or more repeating digits in the decimal part. Examples of terminating decimals are 0.5, 1.4, and 7.9 while non-repeating decimals are 3.1415926…, 1.4142135…., and 2.2360679

Converting repeating decimals into fractions

When working with problems involving repeating decimals, it is important to convert them into simple fractions to maintain accuracy and avoid errors. A repeating decimal can be converted into fraction using the following steps:

EXAMPLE 1

EXAMPLE 2

EXERCFISE 2

Converting fractions into repeating decimals

A fraction can be converted into a decimal by performing a long division. In this process, some fractions will be equivalent to either terminating or non-terminating decimals. If the resulting decimal is a non-terminating with recurring decimals, the repeating digits are indicated by the repeating decimal’s notation.

EXAMPLE 1

EXAMPLE 2

EXERCISE 3

Irrational numbers

Irrational numbers are special numbers in our life. A widely known and used irrational number is Pi (π) which appears in formulas for determining circumferences, areas, and volumes of circular shapes. Activity 2.4 allows you to use your experience in decimals to learn about irrational numbers.

Activity 4: Differentiating types of decimal numbers

1. Identify 10 different fractions which can be converted into terminating or repeating decimals.

2. Use a calculator to find answers to the square roots of at most 10 numbers that have no perfect squares and write your answers with at least 10 decimal places.

3. Compare the answers in tasks 1 and 2 and write down the differences observed.

An irrational number is a number which can be written as a non-terminating and non-repeating decimal. Also, these numbers cannot be expressed in the form of a/b
, where a and b are integers and b ≠ 0.

The set of irrational numbers is denoted by ℚ′. Irrational numbers cannot be represented exactly on a number line. However, they can always be approximated to rational numbers.

Example 1

Note: The numbers such as π, e, eπ, 2π, e + π, and other similar numbers are also called transcendentals.

Example 2

EXERCISE 5

Real numbers

Real numbers are the numbers which include both rational and irrational numbers. A set of real numbers is denoted by ℝ. Thus, all sets of numbers such as natural numbers, whole numbers, integers, rational numbers and irrational numbers are all real numbers.

Natural numbers are the smallest set of numbers followed by whole numbers and integers which are all rational numbers. Rational and irrational numbers are opposite sets of numbers and are what makes the largest set of real numbers as shown in Figure

Note: 1. 0 is also a real number which is neither positive nor negative.

2. Every real number corresponds to a single point on the number line.

3. Every point on the number line corresponds to a certain real number.

Inequalities in real numbers

In daily life, people are faced with problems related to comparing quantities of the same item for the purpose of making decisions. Consider the following examples:

(i) In some places, the speed of the car is limited to a certain maximum value due to the large number of pedestrians.

(ii) Event organisers can set a maximum number of attendees for the event.

(iii) Most banks limit the withdrawal amount to a certain minimum and maximum amount per day in automated teller machines.

(iv) In schools, a minimum pass mark is set for students to be awarded certificates of completion of education.

Activity 2.5 guides you in comparing quantities in real life.

Activity 5: Comparing quantities in real life

1. Measure the lengths, widths, and heights of different objects of your choice.

2. Compare the measured values of a pair of objects in task 1.

In Mathematics, an inequality is a statement that compares or relates two values or expressions. The common terms involved in comparing the values or expressions are “less than”, “greater than”, “greater than or equal to”, “less than or equal to”, or “not equal to”. For instance, the following statements are examples of inequalities:

(i) −5 is less than 0.

(ii) John’s age is greater than Jane’s age.

(iii) x is less than or equal to 9.

(iv) y is greater than or equal to 0.

(v) 2.5 is not equal to 1.2.

The mathematical statements can be written using symbols which represent inequalities as shown in the following table.

In general, mathematical statements which use ≠, >, < ,≥, or ≤ are called inequalities.

Note:

1. The sharp end of > and < always points to a smaller number.

2. The symbols ≤ and ≥ are sometimes referred to as at most and at least, respectively.

3. If the inequality is divided or multiplied both sides by a negative number, the direction of the inequality sign changes.

Suppose a, b, and c are real numbers which are represented on the number line shown in the following number line:

Generally, on a number line, a number to the right side of another number is always greater than a number to its left side. Thus, from the number line,

b is greater than a, which can also be written as b > a.

c is greater than b, which can also be written as c > b.

The vice versa is also true, that is, a number which is on the left side of another number on a number line is always less than the number on its right side. That is, a < b and b < c.

EXAMPLE 1