ICSE Class 8 Maths Chapter 03 Number Systems

Chapter 3

Number Systems

You are already familiar with the system of natural numbers, whole numbers, integers and the four fundamental operations of arithmetic on them - addition, subtraction, multiplication and division. In this chapter, we shall review the same and extend our study to the system of rational numbers, irrational numbers and real numbers.

Natural Numbers

The counting numbers 1, 2, 3, 4, ... are called natural numbers. The set of natural numbers is denoted by N. Thus,

\[N = \{1, 2, 3, 4, ...\}\]

Whole Numbers

The number 0 together with the natural numbers i.e. the numbers 0, 1, 2, 3, 4,... are called whole numbers. The set of whole numbers is denoted by W. Thus,

\[W = \{0, 1, 2, 3, 4, ...\}\]

i.e. \[W = N \cup \{0\}\]

We already know the four fundamental operations of addition, subtraction, multiplication and division on the whole numbers. In particular,

if a, b ≠ 0 are any two whole numbers, then there exist unique whole numbers q and r such that

\[a = b \times q + r \text{ where } 0 \leq r < b\]

i.e.

\[\text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder}\]

This is called division algorithm or division rule.

Integers

We know that the set of whole numbers is closed with respect to the operations of addition and multiplication. However, the set of whole numbers is not closed with respect to the operation of subtraction.

For example, when we subtract 57 from 39 we do not get a whole number i.e. 39 - 57 is not a whole number. So, the system of whole numbers is inadequate for subtraction. Thus, to provide an answer to all the problems of subtraction, we have to enlarge the whole number system. We introduce another kind of numbers called negative integers

i.e. -1, -2, -3, -4, ...

The set of whole numbers together with negative integers is called set of integers. It is denoted by I or Z. Thus,

\[Z = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}\]

All positive and all negative integers including zero are called directed numbers.

Teacher's Note

Understanding whole numbers and integers helps students grasp how transactions work - earning money (positive) and spending it (negative) are natural applications of integers in daily life.

Rules of Calculations

The set of integers is a set of directed numbers. You already know the four fundamental operations of addition, subtraction, multiplication and division on the set of integers. To simplify expressions involving integers, you need to know the rules of calculations. The order in which several operations must be done can be remembered with the help of the word BODMAS.

B - Brackets. First carry out the operation inside brackets.

O - Of. Change 'of' to 'x' and work it out.

D - Division. After 'of', carry out division.

M - Multiplication. After division, carry out multiplication.

A - Addition. After multiplication, carry out addition.

S - Subtraction. Finally carry out subtraction.

According to the rule of BODMAS, calculations must be done in order of the letters in this word.

Brackets

The brackets are the grouping symbols. The different kinds of brackets are:

(i) [ ] are known as rectangular brackets or big brackets.

(ii) { } are known as braces or curly brackets.

(iii) ( ) are known as parenthesis or common brackets.

(iv) - is known as line or bar bracket or vinculum.

The order of removing brackets is:

(i) line bracket (ii) common brackets

(iii) curly brackets and lastly (iv) rectangular brackets.

Example 1

Write whole numbers between 5 to 85 using the digits 8, 0, 6. Repetition of digits is

(i) not allowed (ii) allowed.

Solution

The whole numbers between 5 to 85 consist of one digit or two digits.

The one digit whole numbers formed by the given digits are 0, 6, 8.

From the given digits, the possible ways of choosing two digits are

0, 6; 0, 8; 6, 8

Remember that the digit 0 can not be put at ten's place because that would make the number only one-digited.

(i) When the repetition of digits is not allowed, two-digited whole number formed by the given digits are

60, 80, 68, 86.

Therefore, the whole numbers between 5 to 85 formed by the given digits are

6, 8, 60, 68, 80.

(ii) When the repetition of digits is allowed, two-digited whole numbers formed by the given digits are

60, 66, 80, 88, 68, 86.

Therefore, the whole numbers between 5 to 85 formed by the given digits are

6, 8, 60, 66, 68, 80.

Teacher's Note

Understanding digit combinations helps students recognize patterns in number formation, which is useful in understanding postal codes, phone numbers, and other real-world numerical systems.

Examples and Solutions

Example 2

Write (i) the smallest (ii) the greatest seven-digit whole number having four different digits.

Solution

(i) The smallest four different digits are 0, 1, 2, 3. The smallest seven-digit whole number having four different digits = 1000023

(ii) The greatest four different digits are 9, 8, 7, 6 The greatest seven-digit number having four different digits = 9999876

Example 3

Write the greatest and the smallest five-digit numbers using the digits 0, 2, 5, 7 with the condition that one digit is repeated twice.

Solution

The digits to be used are 0, 2, 5, 7 and one digit is to be used twice.

Greatest number = 77520

Smallest number = 20057

Example 4

Write the greatest and the smallest six-digit numbers using four different digits with the condition that 5 occurs at thousand's place.

Solution

Greatest number = 995987

Smallest number = 105002

Example 5

Find the largest four-digit natural number which is exactly divisible by 439.

Solution

The largest four-digit natural number = 9999. We divide 9999 by 439 and find the remainder.

Therefore, the least number which should be subtracted from 9999 so that the remaining number is exactly divisible by 439 is 341.

Hence, the required number = 9999 - 341 = 9658.

Example 6

Find the smallest five-digit whole number which is exactly divisible by 657.

Solution

The smallest five-digit whole number = 10000. We divide 10000 by 657 and find the remainder.

Therefore, the least number which should be added to 10000 so that the sum is exactly divisible by 657 = 657 - 145 = 512.

Hence, the smallest five-digit whole number which is exactly divisible by 657 = 10000 + 512 = 10512.

Example 7

Simplify the following:

(i) 13 - [3 of (-2) - 18 ÷ (5 - 11 - 4)]

(ii) (-3)³ - 6 ÷ (-3) + 7 of (-5).

Teacher's Note

Teaching BODMAS through real problems helps students organize complex calculations systematically, a skill applicable when budgeting, calculating discounts, or solving multi-step cooking recipes.

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