ICSE Class 7 Maths Chapter 01 Number System

Unit - I: Pure Arithmetic

Chapter 1: Number System

1.1 Review

Teacher's Note

Understanding natural and whole numbers is fundamental to all mathematics. In daily life, we count objects starting from 1 (natural numbers) or from 0 (whole numbers), such as counting apples in a basket or scoring points in a game.

1.2 More About Numbers

1. Prime Numbers (P)

1. Prime numbers are whole numbers greater than 1 and each of which is divisible by unity (1) and by itself only.

2. Except 2, all other prime numbers are odd.

Therefore, Prime numbers, P = 2, 3, 5, 7, 11, 13, ............, etc.

2. Composite Numbers (C)

Composite numbers are whole numbers greater than 1 and none of these numbers is a prime number.

For example: 6 is a whole number greater than 1 and 6 is not a prime number, therefore 6 is a composite number.

Therefore, Composite numbers, C = 4, 6, 8, 9, ................, etc.

The whole number 1 (unity) is neither a prime number nor a composite number.

Example 1

State, with reason, which of the numbers 13 and 18 is a prime number and which is a composite number.

Solution

13 is a prime number (Ans.)

Reason: 13 is a whole number greater than 1(one) and is divisible by 1 and by itself only.

18 is a composite number (Ans.)

Reason: 18 is a whole number greater than 1(one) and is divisible by each of 1, 2, 3, 6, 9 and 18.

Every composite number is divisible by more than two natural numbers.

Teacher's Note

Prime numbers are like the building blocks of all numbers - understanding them helps us factor numbers correctly. In real life, prime numbers are used in computer encryption to keep our online transactions secure.

3. Integers (Z or I)

1. The integers consist of natural numbers, zero and negative of natural numbers.

Thus, Z or I = ............, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ..................

2. There are infinite integers towards positive side and infinite integers towards negative side.

3. Positive Integers, (Z+) = 1, 2, 3, 4, ........., etc. = N

Negative integers, (Z-) = -1, -2, -3, ......, etc.

4. Rational Numbers (Q)

The numbers, each of which can be expressed in the form \(\frac{a}{b}\), where a and b both are integers and b is not equal to zero, are called rational numbers.

i.e. \(\frac{a}{b}\) is a rational number when; a, b ∈ Z and b ≠ 0.

For example:

(i) \(\frac{2}{5}\) is a rational number, since 2, 5 ∈ Z and 5 ≠ 0.

(ii) 5 is a rational number, since, 5 = \(\frac{5}{1}\), where 5, 1 ∈ Z and 1 ≠ 0.

(iii) 0 is a rational number, as 0 can be expressed as \(\frac{0}{5}, \frac{0}{-7}\), etc.

(iv) \(\frac{5}{0}\) is not a rational number, since 5, 0 ∈ Z but denominator = 0.

(v) \(\sqrt{2}, \sqrt{3}, \sqrt{5}\), etc., are not rational numbers, since these numbers can not be expressed as \(\frac{a}{b}\), where a, b ∈ Z and b ≠ 0.

(vi) \(-\frac{3}{4}\) is a rational number, since -3, 4 ∈ Z and 4 ≠ 0.

Remember: \(\frac{3}{4} = \frac{-3}{-4} = \frac{3}{-4}\)

(vii) 0.3 is a rational number, since 0.3 = \(\frac{3}{10}\); where 3, 10 ∈ Z and 10 ≠ 0.

1. Each natural number, each whole number, each integer and each fraction (including decimal fraction) is a rational number.

2. There are infinite number of rational numbers.

Teacher's Note

Rational numbers help us represent parts of a whole, which we use constantly in cooking (half a cup of flour), shopping (dividing the cost), and measuring distances.

5. Irrational Numbers (Q̄)

The numbers, which are not rational, are called irrational numbers.

Each of -\(\sqrt{7}\), -\(\sqrt{2}\), 3\(\sqrt{4}\), \(\sqrt{5}\), etc., is an irrational number.

Note: The number \(\frac{a}{b}\) is neither rational nor irrational, if b = 0; e.g., \(\frac{5}{0}, \frac{7}{0}, \frac{-3}{0}\), etc.

Infact, if b = 0, \(\frac{a}{b}\) is said to be not defined or infinity.

6. Real Numbers (R)

Numbers which are either rational or irrational are called real numbers.

(i) Each natural number is a real number,

(ii) Each whole number is a real number,

(iii) Each integer is a real number,

(iv) Each rational number is a real number,

(v) Each irrational number is also a real number.

1.3 Number Line

A number line is shown below:

[Visual representation of number line from -4 to 5]

1. Arrow-heads are drawn at both the ends of the line to show that the line continues to be of infinite length and so the numbers are also upto infinity on both the sides.

2. To the left of zero, on this line, are negative and to its right are positive numbers.

3. For any two numbers on the number line, the one which is on the right of the other, is greater. And, the number, which is on the left, is smaller.

For example: (i) -1 is to the right of -2, therefore -1 is greater than -2

(ii) -1 is to the left of 3, therefore -1 is smaller than 3 and so on.

1. If a is greater than b; then -a is smaller than -b, i.e., if a > b; then -a < -b.

And, if a is smaller than b; then -a is greater than -b. i.e., if a < b; then -a > -b.

For example: (i) 8 > 5; but -8 < -5.

(ii) 2 < 6; but -2 > -6.

(iii) -3 > -6; but 3 < 6.

(iv) -15 < -8; but 15 > 8.

2. (i) Every negative number is smaller than zero (0).

(ii) Every positive number is greater than zero (0).

(iii) Every positive number is greater than every negative number.

Teacher's Note

The number line helps visualize temperature changes (degrees above and below zero), altitude (heights above and below sea level), and bank account balances (gains and debits).

1.4 The Face and the Place Values of a Digit

Each and every number (whether it is very small or very-very large) is made up of one or more of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Each of these ten symbols is called a digit.

For example:

(a) Each of 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 is a single digit number.

(b) Each of 10, 22, 53, 78, 87, 90, etc., is a two digit number.

(c) Each of 3297, 5228, 4444, etc., is a four digit number and so on.

1. The face value of a digit

In a given number, the face (true) value of a digit is the digit itself.

For example:

In the number 837;

the face value of 8 is 8, the face value of 3 is 3 and the face value of 7 is 7.

2. The place value of a digit

In a given number, the place (local) value of a non-zero digit depends on the place it occupies in the given number.

For example:

In the number 837;

the digit 8 occupies hundred's place, so its place value is 8 × 100 = 800;

the digit 3 occupies ten's place, so its place value is 3 × 10 = 30

and the digit 7 occupies unit's place, so its place value is 7 × 1 = 7.

Example 2

Write all possible 2-digit numbers formed out of the digits 4, 2 and 9, if:

(i) repetition of digits is allowed.

(ii) repetition of digits is not allowed.

Solution

(i) Required 2-digit numbers = 44, 42, 49, 24, 22, 29, 94, 92 and 99. (Ans.)

(ii) Required numbers = 42, 49, 24, 29, 94 and 92 (Ans.)

Example 3

Write all possible 3-digit numbers formed out of the digits 4, 2 and 9, if:

Repetition of digits is not allowed.

Solution

Required numbers = 429, 492, 249, 294, 924 and 942. (Ans.)

Teacher's Note

Understanding place value is crucial for performing arithmetic operations correctly. In real life, this applies to reading large numbers like population statistics, distances between planets, or the cost of properties.

This is a preview of the first 3 pages. To get the complete book, click below.