Exercise 1.1

1. Is zero a rational number? Can you write it in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0?

Solution:

Yes, Zero is a rational Number as it can be written in the form of \(\frac{p}{q}| q\ne 0\) e.g. \(\frac{0}{1} = \frac{0}{2} = \frac{0}{3} = \frac{0}{4} = \cdots\) Denominators may be taken as negative numbers.

2. Find six rational numbers between 3 and 4

Solution

To solve such a question, the easiest way is to take any number greater than the required rational number, which is 6 in this case. we can take 6 + 1 = 7; multiply and divide 3 and 4 by 7 \[3=\left( 3\times \frac{7}{7}\right) \text{ and 4 =} \left( 4\times \frac{7}{7}\right)\] \[3=\left(\frac{21}{7}\right) \text{ and 4 =} \left(\frac{28}{7}\right)\] We need to find 6 rational numbers between these \[\frac{21}{7},\dots\cdots,\frac{28}{7}\] which can be obtained by incrementing the numerator by one in each number \[\color{blue}{\frac{21}{7}},\color{red}{\frac{22}{7},\frac{23}{7},\frac{24}{7},\frac{25}{7},\frac{26}{7}, \frac{27}{7}},\color{blue}{\frac{28}{7}}\] Numbers in red are the required Rationl numbers between 3 & 4.

3. Find five rational numbers between \(\frac{3}{5} \text{ and }\frac{4}{5}\).

Solution

Numbers to find between \(\frac{3}{5} \text{ and }\frac{4}{5}\) = 5\(\text{therefore we will take a number } \ge \text{5 i.e. 5 + 1 = 6}\) Multiply and divide \(\frac{3}{5} \text{ and }\frac{4}{5}\) by 6. $$\begin{aligned}\dfrac{3}{5}\times \dfrac{6}{6}=\dfrac{18}{30}\\\dfrac{4}{5}\times \dfrac{6}{6}=\dfrac{24}{30}\end{aligned}$$ To find 5 rational Numbers between Increment the numerator by 1 till it reaches to 23 $$\dfrac{18}{30},\cdots\cdots, \dfrac{24}{30}$$ $$\color{blue}{\dfrac{18}{30},}\color{red}{\dfrac{19}{30},\dfrac{20}{30},\dfrac{21}{30},\dfrac{22}{30},\dfrac{23}{30},}\color{blue}{\dfrac{24}{30}}$$ 5 Rational Numbers in red are the required numbers.

4. State whether the following statements are true or false. Give reasons for your answers.

Solution:

1. Every natural number is a whole number. True
2. Every integer is a whole number. False (-1 is an integer bot not a whole number)
3. Every rational number is a whole number. False \(\frac{1}{2}\) is a Rational Number but not a whole number