ICSE Class 9 Maths Chapter 01 Rational and Irrational Numbers

Unit 1: Pure Arithmetic

Rational and Irrational Numbers

Introduction

The complete number system is divided into two types of numbers:

1. Imaginary numbers - 2. Real numbers

For example:

1. If \(x = 4\), \(\sqrt{-x}\) i.e. \(\sqrt{-4}\) is an imaginary number and \(\sqrt{x} = \sqrt{4} = 2\) is a real number.

2. \(\sqrt{-5}\) is imaginary but \(\sqrt{5}\) is real and so on.

Thus, square root of every negative number is an imaginary number and if the number is not imaginary, it is a real number. In this chapter, we confine our studies only upto real numbers.

Starting from real numbers, the complete number system is as shown below:

Real Numbers (R)

Rational Numbers (Q): \(-8, 0, 6, \frac{5}{8}, \frac{2}{9}\), etc. - Irrational Numbers (\(\bar{Q}\)): \(\sqrt{5}, \sqrt{3}, -\sqrt{8}, 3\sqrt{2}, 2, -\sqrt{7}\), etc.

Integers (I or Z): \(..., -2, -1, 0, 1, 2, ...\) - Non-Integral Rationals: \(\frac{5}{8}, \frac{8}{15}, \frac{2}{7}\), etc.

Negative Integers: \(..., -3, -2, -1\) - Zero: \(\{0\}\) - Positive Integers (Natural Numbers, N): \(\{1, 2, 3, 4, ...\}\)

Whole-Numbers (W): \(\{0, 1, 2, 3, 4, 5, etc.\}\)

Rational Numbers (Q)

A number which can be expressed as \(\frac{a}{b}\), where 'a' and 'b' both are integers and 'b' is not equal to zero, is called a rational number.

In general, the set of rational numbers is denoted by the letter Q.

\(\therefore Q = \{\frac{a}{b} : a, b \in Z \text{ and } b \neq 0\}\)

Properties of Rational Numbers

1. \(\frac{a}{b}\) is a rational number

\(\Rightarrow\) (i) \(b \neq 0\)

(ii) a and b have no common factor other than 1 (one) i.e. a and b are co-primes.

(iii) b is usually positive, whereas a may be positive, negative or zero.

2. Every integer (positive, negative or zero) and every decimal number is a rational number.

3. Corresponding to every rational number \(\frac{a}{b}\), its negative rational number is \(\frac{-a}{b}\).

Also, \(\frac{-a}{b} = \frac{a}{-b} = -\frac{a}{b}\) e.g. \(\frac{-3}{5} = \frac{3}{-5} = -\frac{3}{5}\) and so on.

4. Two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\) are equal, if and only if: \(a \times d = b \times c\).

i.e. \(\frac{a}{b} = \frac{c}{d} \Leftrightarrow a \times d = b \times c\)

Also, \(\frac{a}{b} > \frac{c}{d} \Leftrightarrow a \times d > b \times c\) and \(\frac{a}{b} < \frac{c}{d} \Leftrightarrow a \times d < b \times c\).

5. For any two rational numbers a and b, \(\frac{a+b}{2}\) is also a rational number which lies between a and b. Thus:

if \(a > b \Rightarrow a > \frac{a+b}{2} > b\) and if \(a < b \Rightarrow a < \frac{a+b}{2} < b\).

Example 1

Insert three rational numbers between 3 and 5.

Solution

Since, \(3 < 5 \Rightarrow 3 < \frac{3+5}{2} < 5\).

(Inserting one rational number between 3 and 5)

\(\Rightarrow 3 < 4 < 5\)

\(\Rightarrow 3 < \frac{3+4}{2} < 4 < \frac{4+5}{2} < 5\)

\(\Rightarrow 3 < \frac{7}{2} < 4 < \frac{9}{2} < 5 \Rightarrow 3 < 3\frac{1}{2} < 4 < 4\frac{1}{2} < 5\)

\(\therefore 3\frac{1}{2}, 4 \text{ and } 4\frac{1}{2}\) are three rational numbers between 3 and 5.

Teacher's Note

Understanding rational numbers is like dividing a pizza into fair portions - we need to know how many pieces (numerator) and the total number of cuts (denominator) to represent any fraction accurately.

Key Points on Rational Numbers

1. There are infinitely many rational numbers between each pair of rational numbers.

2. For rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\), \(\frac{a+c}{b+d}\) is also a rational number with its value lying between \(\frac{a}{b}\) and \(\frac{c}{d}\).

Example 1 (continued)

For example 1, given above:

Consider the rational numbers 3 and 5

i.e. \(\frac{3}{1}\) and \(\frac{5}{1}\) where \(\frac{3}{1} < \frac{5}{1}\)

\(\Rightarrow \frac{3}{1} < \frac{3+5}{1+1} < \frac{5}{1}\) i.e. \(\frac{3}{1} < \frac{4}{1} < \frac{5}{1}\)

\(\Rightarrow \frac{3}{1} < \frac{3+4}{1+1} < \frac{4}{1} < \frac{4+5}{1+1} < \frac{5}{1}\)

\(\Rightarrow 3 < \frac{7}{2} < 4 < \frac{9}{2} < 5\) i.e. \(3 < 3\frac{1}{2} < 4 < 4\frac{1}{2} < 5\)

Also, every terminating and non-terminating recurring decimal number between 3 and 5 is a rational number between 3 and 5.

For example:

(i) \(3.2 < 3.85 < 4.3\) - (ii) \(4.97 > 4.294 > 3.87 > 3.2\)

Method For Finding Large Number of Rational Numbers Between Two Given Rational Numbers

Let x and y be two rational numbers such that \(x < y\).

In order to find n rational numbers between x and y, first of all find \(d = \frac{y-x}{n+1}\)

Then, n rational number between x and y are:

\(x + d, x + 2d, x + 3d, ..., x + nd\).

In example 1, given above: \(3 < 5\)

\(\Rightarrow x = 3\) and \(y = 5\)

To insert 3 rational numbers between 3 and 5, \(n = 3\)

\(\Rightarrow d = \frac{y-x}{n+1} = \frac{5-3}{3+1} = \frac{2}{4} = \frac{1}{2}\).

\(\therefore\) Required rational numbers are: \(x + d, x + 2d \text{ and } x + 3d\)

\(= 3 + \frac{1}{2}, 3 + 2 \times \frac{1}{2} \text{ and } 3 + 3 \times \frac{1}{2} = 3\frac{1}{2}, 4 \text{ and } 4\frac{1}{2}\)

Example 2

Find four rational numbers between \(\frac{2}{3}\) and \(\frac{5}{6}\).

Solution

Since, \(\frac{2}{3} < \frac{5}{6}\)

(As, \(2 \times 6 < 5 \times 3\))

\(\Rightarrow x = \frac{2}{3}, y = \frac{5}{6} \text{ and } n = 4\)

\(\therefore d = \frac{y-x}{n+1} = \frac{\frac{5}{6} - \frac{2}{3}}{4+1} = \frac{\frac{5-2 \times 2}{6}}{5} = \frac{5-4}{5 \times 6} = \frac{1}{30}\)

\(\Rightarrow\) Required rational numbers are:

\(= x + d, x + 2d, x + 3d \text{ and } x + 4d\)

\(= \frac{2}{3} + \frac{1}{30}, \frac{2}{3} + 2 \times \frac{1}{30}, \frac{2}{3} + 3 \times \frac{1}{30} \text{ and } \frac{2}{3} + 4 \times \frac{1}{30}\)

\(= \frac{20}{30} + \frac{1}{30}, \frac{20}{30} + \frac{2}{30}, \frac{20}{30} + \frac{3}{30} \text{ and } \frac{20}{30} + \frac{4}{30}\)

\(= \frac{21}{30}, \frac{22}{30}, \frac{23}{30} \text{ and } \frac{24}{30} = \frac{7}{10}, \frac{11}{15}, \frac{23}{30} \text{ and } \frac{4}{5}\)

Alternative method

For finding 4 rational numbers between \(\frac{2}{3}\) and \(\frac{5}{6}\):

1. Find L.C.M. of the denominators. L.C.M. of denominators 3 and 6 = 6.

2. Make denominator of each given rational number equal to 6 (the L.C.M.).

\(\therefore \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \text{ and } \frac{5}{6} = \frac{5}{6}\).

3. Since, 4 rational numbers are required, multiply the numerator and denominator of each rational number (obtained in step 2) by \(4 + 1 = 5\).

\(\therefore \frac{4}{6} = \frac{4 \times 5}{6 \times 5} = \frac{20}{30} \text{ and } \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}\).

Now, every rational number with denominator 30 and numerator between 20 and 25 will have its value between the given rational numbers \(\frac{2}{3}\) and \(\frac{5}{6}\).

\(\Rightarrow\) Required rational numbers between \(\frac{2}{3}\) and \(\frac{5}{6}\) are:

\(= \frac{21}{30}, \frac{22}{30}, \frac{23}{30} \text{ and } \frac{24}{30} = \frac{7}{10}, \frac{11}{15}, \frac{23}{30} \text{ and } \frac{4}{5}\)

Example 3

Insert three rational numbers between 2.6 and 3.1.

Solution

First method:

\(2.6 < 3.1 \Rightarrow 2.6 < \frac{2.6+3.1}{2} < 3.1\)

\(\Rightarrow 2.6 < 2.85 < 3.1\)

\(\Rightarrow 2.6 < \frac{2.6+2.85}{2} < 2.85 < \frac{2.85+3.1}{2} < 3.1\)

\(\Rightarrow 2.6 < 2.725 < 2.85 < 2.975 < 3.1\)

\(\therefore\) Required rational numbers are: 2.725, 2.85 and 2.975

Second method:

\(2.6 < 3.1 \Rightarrow \frac{26}{10} < \frac{31}{10}\)

\(\Rightarrow \frac{26}{10} < \frac{26+31}{10+10} < \frac{31}{10}\)

\(\Rightarrow \frac{26}{10} < \frac{57}{20} < \frac{31}{10}\)

\(\Rightarrow \frac{26}{10} < \frac{26+57}{10+20} < \frac{57}{20} < \frac{57+31}{20+10} < \frac{31}{10}\)

\(\Rightarrow \frac{26}{10} < \frac{83}{30} < \frac{57}{20} < \frac{88}{30} < \frac{31}{10}\)

\(\Rightarrow 2.6 < 2.77 < 2.85 < 2.93 < 3.1\)

\(\therefore\) Required rational numbers are: 2.77, 2.85 and 2.93

Third method:

Since, \(2.6 < 3.1\), therefore let \(x = 2.6\) and \(y = 3.1\). Also, \(n = 3\)

\(\therefore d = \frac{y-x}{n+1} = \frac{3.1-2.6}{3+1} = \frac{0.5}{4} = 0.125 = 0.13\) (Correct to two decimal places)

\(\Rightarrow\) Required rational numbers are:

\(x + d, x + 2d \text{ and } x + 3d\)

\(= 2.6 + 0.13, 2.6 + 2 \times 0.13 \text{ and } 2.6 + 3 \times 0.13 = 2.73, 2.86 \text{ and } 2.99\)

Fourth method:

Since, \(2.6 < 3.1\), therefore let \(x = 2.6\) and \(y = 3.1\)

\(2.6, 3.1 = \frac{26}{10}, \frac{31}{10}\)

\(= \frac{26 \times 4}{10 \times 4}, \frac{31 \times 4}{10 \times 4}\)

\(= \frac{104}{40}, \frac{124}{40}\)

(\(\therefore n + 1 = 3 + 1 = 4\))

Now, every rational number with denominator 40 and numerator between 104 and 124 will lie between given rational numbers 2.6 and 3.1.

\(\therefore\) Required rational numbers can be taken as:

\(\frac{106}{40}, \frac{110}{40} \text{ and } \frac{120}{40} = 2.65, 2.75 \text{ and } 3\)

Example 4

Which of the rational numbers \(\frac{3}{5}\) and \(\frac{5}{7}\) is greater. Insert three rational numbers between \(\frac{3}{5}\) and \(\frac{5}{7}\) so that all the five numbers are in ascending order of their values.

Solution

\(\frac{3}{5} \text{ and } \frac{5}{7} = \frac{3 \times 7}{5 \times 7} \text{ and } \frac{5 \times 5}{7 \times 5} = \frac{21}{35} \text{ and } \frac{25}{35}\)

(L.C.M. of 5 and 7 = 35)

Since, \(21 < 25 \Rightarrow \frac{21}{35} < \frac{25}{35} \Rightarrow \frac{3}{5} < \frac{5}{7} \Rightarrow \frac{5}{7}\) is greater.

Now, \(\frac{3}{5} < \frac{5}{7} \Rightarrow \frac{3}{5}, \frac{\frac{3}{5}+\frac{5}{7}}{2} < \frac{5}{7}\)

\(\Rightarrow \frac{3}{5} < \frac{23}{35} < \frac{5}{7}\)

[\(\frac{\frac{3}{5}+\frac{5}{7}}{2} = \frac{21+25}{2 \times 35} = \frac{46}{70} = \frac{23}{35}\)]

\(\Rightarrow \frac{3}{5} < \frac{\frac{3}{5}+\frac{23}{35}}{2} < \frac{23}{35} < \frac{\frac{23}{35}+\frac{5}{7}}{2} < \frac{5}{7}\)

\(\Rightarrow \frac{3}{5} < \frac{22}{35} < \frac{23}{35} < \frac{24}{35} < \frac{5}{7}\)

Which are in ascending order of their values.

Teacher's Note

Organizing numbers in order is like arranging students by height - we need a systematic method to compare and insert values between any two points on a number scale.

Properties of Rational Numbers (Q)

1. The sum of two or more rational numbers is always a rational number.

2. The difference of two rational numbers is always a rational number. If a and b are any two rational numbers, then each of \(a - b\) and \(b - a\) is also a rational number.

3. The product of two or more rational numbers is always a rational number.

4. The division of a rational number by a non-zero rational number is always a rational number.

If a and b are any two rational numbers and \(b \neq 0\); then \(\frac{a}{b}\) is always a rational number.

Since, the sum (addition) of two rational numbers is always a rational number; we say that the set of rational numbers is closed for addition. In the same way, the set of rational numbers is closed for: (i) subtraction - (ii) multiplication and - (iii) division; if divisor \(\neq 0\).

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