ICSE Class 8 Maths Chapter 01 Number System

Unit - 1: Pure Arithmetic

Chapter 1: Number System

1.1 Review

Natural Numbers (N)

(i) Each of 1, 2, 3, 4, ......., etc., is a natural number.

(ii) The first and the smallest natural number is one (1); whereas the last and the largest natural number cannot be obtained.

(iii) Consecutive natural numbers differ by one (1).

Whole Numbers (W)

(i) Each of 0, 1, 2, 3, 4, ........, etc., is a whole number.

(ii) The first and the smallest whole number is zero (0); whereas the last and the largest whole number cannot be obtained.

(iii) Consecutive whole numbers differ by one (1).

(iv) Except zero (0) every whole number is a natural number.

Integers (I or Z)

(i) Each of ........, -4, -3, -2, -1, 0, 1, 2, 3, 4, ......... is an integer.

(ii) Every integer is either:

(a) negative of a natural number, such as -1, -2, -3, -4, ....., etc.

(b) zero i.e. 0, or

(c) a natural number i.e. 1, 2, 3, 4, ......., etc.

(iii) The smallest (the first) and the largest (the last) integers cannot be obtained.

(iv) Consecutive integers differ by one (1).

Rational Numbers (Q)

(i) It is a number which can be expressed as \(\frac{a}{b}\), where a and b both are integers and b ≠ 0.

(ii) Every fraction such as \(\frac{3}{8}\), \(\frac{7}{15}\), \(-\frac{6}{11}\) etc. is a rational number.

(iii) Since, \(8 = \frac{8}{1}\), \(-3 = \frac{-3}{1}\), \(23 = \frac{23}{1}\), etc., therefore each integer is a rational number. For the same reason, each natural number and each whole number is also a rational number.

(iv) Zero (0) can be written as: \(\frac{0}{5}\), \(\frac{0}{-8}\), \(\frac{0}{10}\), etc., therefore zero is also a rational number.

(v) Every decimal number can be expressed as a fraction; so it is also a rational number.

e.g. \(0.7 = \frac{10}{10}\), \(2.5 = \frac{25}{10} = \frac{5}{2}\), etc.

Irrational Numbers (Q̄)

(i) The number which cannot be expressed as \(\frac{a}{b}\); where a ∈ I, b ∈ I and b ≠ 0; is called an irrational number.

Infact a number, which is not rational, is irrational.

(ii) Each of \(\sqrt{2}\), \(\sqrt{7}\), \(\sqrt{15}\), \(3\sqrt{7}\), \(3 - \sqrt{2}\), \(\sqrt{5} + \sqrt{3}\), etc., is an irrational number.

(iii) Each non-terminating and non-recurring number is an irrational number.

Real Numbers (R)

(i) Rational numbers and irrational numbers, taken together, are called real numbers.

i.e. R = Q ∪ Q̄

(ii) Every real number is either rational or irrational.

Test Yourself

1. 5 is an ..................., 9 is an ................... and \(\frac{5}{9}\) is a ..................... ..............................................

2. -8 is an ...................., 15 is an ................... and \(\frac{-8}{15}\) is a ..................... ..............................................

3. 5 is a ..................... ....................... and \(\sqrt{7}\) is an .................... ............................................., then each of \(5 + \sqrt{7}\), \(5 - \sqrt{7}\), \(\sqrt{7} - 5\) and \(5\sqrt{7}\) is an .................... ..............................................

4. \(\sqrt{13}\) is an .................... ........................................ and 8 is a .................... ....................................... then each of : \(8\sqrt{13}\), \(\sqrt{13} + 8\), \(8 - \sqrt{13}\), etc., is an .................... ..............................................

Teacher's Note

Understanding number systems is fundamental to mathematics, much like understanding the alphabet is essential for reading and writing in any language.

1.2 Prime and Composite Numbers

Prime Number

A natural number, which is greater than 1 and is divisible by one (1) and itself only, is called a prime number.

(i) 5 is a prime number as it is greater than one and is divisible by one (1) and itself only.

For the same reason, each of the following is a prime number:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.

(ii) Two (2) is the smallest prime number and the largest prime number cannot be obtained. In other words, there are an infinite number of prime numbers.

(iii) Except 2, every prime number is an odd natural number.

Composite Number

A natural number, which is greater than 1 and is not prime, is called a composite number.

A composite number is divisible by 1 (one), by itself and atleast by one more number.

6 is divisible by 1, by itself and by 2 and 3 also; therefore 6 is a composite number.

(i) Each of 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc. is a composite number.

(ii) The smallest composite number is 4 and the largest composite number cannot be obtained i.e. there are an infinite number of composite numbers.

(iii) A composite number can be even or odd.

One (1) is neither prime nor composite.

Teacher's Note

Prime numbers are used in encryption for online banking and security - they are the building blocks of digital safety in our connected world.

1.3 More About Numbers

Twin Primes

Prime numbers, which differ by two, are said to be twin prime numbers.

e.g. 5 and 7; 11 and 13, etc.

Prime Triplet

The set of three consecutive prime numbers with a difference of 2 is called the prime triplet. Note that there is only one set {3, 5, 7} of the prime triplet.

Co-primes

The pairs of numbers, which are not divisible by a common number other than 1, are called co-primes.

e.g. 10 and 21 are co-primes, since these numbers cannot be divided by the same number other than 1.

Similarly, 7 and 16; 8 and 25; 16 and 25 are some more pairs of co-primes.

Teacher's Note

Co-prime pairs appear in ratio and proportion problems - they help simplify fractions to their lowest terms in daily calculations.

1.4 Tests of Divisibility

Division by 2, 4 and 8

(i) A number is divisible by 2, if its last (unit) digit is divisible by 2 (e.g. 78, 50, 146, etc.)

(ii) A number is divisible by 4, if the number formed by its last two digits is divisible by 4.

e.g. 172 is divisible by 4 as the number 72 formed by its last two digits is divisible by 4. Similarly, 392, 500, 29320, etc. are also divisible by 4.

A number is divisible by 8, if the number formed by its last three digits is divisible by 8 (e.g. 33176, 436000, etc.)

Division by 3 and 9

(i) A number is divisible by 3, if the sum of its digits is divisible by 3 (e.g. 27, 192, 54924, etc.)

Since the sum of digits of 192 = 1 + 9 + 2 = 12, which is divisible by 3, therefore 192 is divisible by 3.

(ii) A number is divisible by 9, if the sum of its digits is divisible by 9 (e.g. 27, 198, etc.).

Division by 6

If a number is divisible by 3 as well as by 2, it is divisible by 6.

Division by 5 and 10

(i) A number is divisible by 5, if its last (unit) digit is 5 or 0 (e.g. 345, 240, etc.).

(ii) A number is divisible by 10, if its last digit is 0 (e.g. 310, 4000, etc.)

Division by 11

Find the sum of the digits in the even places of the given number and the sum of the digits in its odd places. If the difference between the two sums is 0 or a multiple of 11, then the given number is divisible by 11.

e.g. consider the number 72512. The sum of its digits at even places = 2 + 1 = 3 and the sum of its digits at odd places = 7 + 5 + 2 = 14.

The difference between the two sums = 14 - 3 = 11; which is a multiple of 11.

∴ 72512 is divisible by 11.

In the same way, each of 957, 1496, 68772 is divisible by 11.

Teacher's Note

Divisibility rules are shortcuts used by shopkeepers and accountants to quickly verify calculations and check numbers for accuracy.

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