Number Systems
NCERT Solutions • Class 9 Maths • Chapter 1Exercise 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 $q \neq 0$?
Solution:
Yes, zero is a rational number.
We can write zero in the form $\frac{p}{q}$ as: $$ \frac{0}{1}, \frac{0}{2}, \frac{0}{3}, \frac{0}{-5}, \dots $$ In all these cases, $p = 0$ (which is an integer) and $q$ is a non-zero integer.
We can write zero in the form $\frac{p}{q}$ as: $$ \frac{0}{1}, \frac{0}{2}, \frac{0}{3}, \frac{0}{-5}, \dots $$ In all these cases, $p = 0$ (which is an integer) and $q$ is a non-zero integer.
Note: The denominator $q$ can also be negative, but it cannot be zero.
2. Find six rational numbers between 3 and 4.
Solution:
To find 6 rational numbers, we can multiply and divide both numbers by $6 + 1 = 7$.
Step 1: Convert 3 and 4 into rational numbers with denominator 7. $$ 3 = \frac{3 \times 7}{1 \times 7} = \frac{21}{7} $$ $$ 4 = \frac{4 \times 7}{1 \times 7} = \frac{28}{7} $$
Step 2: Choose numbers between the numerators 21 and 28. The integers between 21 and 28 are 22, 23, 24, 25, 26, and 27.
Answer: The six rational numbers are: $$ \frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7} $$
Step 1: Convert 3 and 4 into rational numbers with denominator 7. $$ 3 = \frac{3 \times 7}{1 \times 7} = \frac{21}{7} $$ $$ 4 = \frac{4 \times 7}{1 \times 7} = \frac{28}{7} $$
Step 2: Choose numbers between the numerators 21 and 28. The integers between 21 and 28 are 22, 23, 24, 25, 26, and 27.
Answer: The six rational numbers are: $$ \frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7} $$
3. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.
Solution:
To find 5 rational numbers, we multiply the numerator and denominator by $5 + 1 = 6$ (or any number larger than 5).
Step 1: Multiply by 6. $$ \frac{3 \times 6}{5 \times 6} = \frac{18}{30} $$ $$ \frac{4 \times 6}{5 \times 6} = \frac{24}{30} $$
Step 2: Choose numbers between the numerators 18 and 24. The integers are 19, 20, 21, 22, and 23.
Answer: The five rational numbers are: $$ \frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30} $$
Step 1: Multiply by 6. $$ \frac{3 \times 6}{5 \times 6} = \frac{18}{30} $$ $$ \frac{4 \times 6}{5 \times 6} = \frac{24}{30} $$
Step 2: Choose numbers between the numerators 18 and 24. The integers are 19, 20, 21, 22, and 23.
Answer: The five rational numbers are: $$ \frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30} $$
4. State whether the following statements are true or false. Give reasons for your answers.
Solution:
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(i) Every natural number is a whole number.
True.Reason: Natural numbers ($1, 2, 3, \dots$) are a part of the collection of Whole numbers ($0, 1, 2, 3, \dots$). -
(ii) Every integer is a whole number.
False.Reason: Integers include negative numbers (like $-2, -5$), which are not Whole numbers. Whole numbers are only from $0$ to positive infinity. -
(iii) Every rational number is a whole number.
False.Reason: Rational numbers include fractions (like $\frac{1}{2}, \frac{2}{3}$) which are not Whole numbers.