Chapter 1: Real Numbers

Chapter 1: Real Numbers (Exercise 1.1)

Question 1:

Write any three rational numbers.

(कोई तीन परिमेय संख्याएँ लिखिए।)

Solution:

A number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$, is called a rational number.

Three rational numbers are:

1. $\frac{2}{3}$

2. $-\frac{5}{7}$

3. $4$ (since $4 = \frac{4}{1}$)

Question 2:

Find five rational numbers between $1$ and $2$.

($1$ और $2$ के बीच पाँच परिमेय संख्याएँ ज्ञात कीजिए।)

Solution:

To find $5$ rational numbers, we multiply the numerator and denominator of both numbers by $5 + 1 = 6$.

Converting $1$ and $2$ into rational numbers with denominator $6$:

$$1 = \frac{1 \times 6}{1 \times 6} = \frac{6}{6}$$

$$2 = \frac{2 \times 6}{1 \times 6} = \frac{12}{6}$$

The rational numbers between $\frac{6}{6}$ and $\frac{12}{6}$ are:

$$\frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}, \text{ and } \frac{11}{6}$$

Question 3:

Express the following in the form of $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$:

1. $0.36$

2. $15.4$

Solution:

1. For $0.36$:

$$0.36 = \frac{36}{100}$$

Dividing both numerator and denominator by $4$:

$$0.36 = \frac{9}{25}$$

2. For $15.4$:

$$15.4 = \frac{154}{10}$$

Dividing both numerator and denominator by $2$:

$$15.4 = \frac{77}{5}$$

Question 4:

Express $0.\overline{6}$ in $\frac{p}{q}$ form.

($0.\overline{6}$ को $\frac{p}{q}$ रूप में व्यक्त कीजिए।)

Solution:

Let $x = 0.6666...$ --- (Equation 1)

Since $1$ digit is repeating, multiply both sides by $10$:

$$10 \times x = 10 \times 0.6666...$$

$$10x = 6.6666...$$

--- (Equation 2)

Subtracting Equation 1 from Equation 2:

$$10x - x = (6.6666...) - (0.6666...)$$

$$9x = 6$$

$$x = \frac{6}{9} = \frac{2}{3}$$

Therefore, $0.\overline{6} = \frac{2}{3}$.