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Chapter 1: Number Systems

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This page provides comprehensive Chapter 1: Number Systems - Basic Worksheet - SJMaths. Basic level practice worksheet for Class 9 Number Systems.

Basic Level Worksheet

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  1. Question 1: Is zero a rational number? Can you write it in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$?
    Solution:
    Yes, zero is a rational number.
    It can be written in the form $\frac{p}{q}$ as $\frac{0}{1}, \frac{0}{2}, \frac{0}{-5}$, etc., where the numerator $p=0$ is an integer and the denominator $q \neq 0$ is an integer.
  2. Question 2: Find six rational numbers between 3 and 4.
    Solution:
    To find 6 rational numbers, we multiply and divide 3 and 4 by $6+1=7$.
    $3 = \frac{3 \times 7}{7} = \frac{21}{7}$ and $4 = \frac{4 \times 7}{7} = \frac{28}{7}$.
    The six rational numbers are: $\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}$.
  3. Question 3: Locate $\sqrt{2}$ on the number line.
    Solution:
    1. Draw a number line and mark point O at 0 and A at 1.
    2. At A, draw a perpendicular unit length AB = 1 unit.
    3. Join OB. By Pythagoras theorem, $OB^2 = OA^2 + AB^2 = 1^2 + 1^2 = 2 \Rightarrow OB = \sqrt{2}$.
    4. With O as center and radius OB, draw an arc cutting the number line at P. Point P represents $\sqrt{2}$.
  4. Question 4: Express $0.4\overline{7}$ in the form $p/q$.
    Solution:
    Let $x = 0.4777\dots$ (i)
    Multiply by 10: $10x = 4.777\dots$ (ii)
    Multiply by 100: $100x = 47.777\dots$ (iii)
    Subtract (ii) from (iii): $90x = 43 \Rightarrow x = \frac{43}{90}$.
  5. Question 5: Rationalize the denominator of $\frac{1}{\sqrt{7}}$.
    Solution: $\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7}$.
  6. Question 6: Find the decimal expansion of $\frac{10}{3}$.
    Solution: Dividing 10 by 3 gives $3.333\dots$. It is a non-terminating repeating decimal ($3.\overline{3}$).
  7. Question 7: Is $\pi$ (pi) rational or irrational?
    Solution: $\pi$ is an irrational number because its decimal expansion is non-terminating and non-recurring.
  8. Question 8: Simplify: $(\sqrt{5} + \sqrt{2})^2$.
    Solution: Using $(a+b)^2 = a^2 + b^2 + 2ab$:
    $(\sqrt{5})^2 + (\sqrt{2})^2 + 2(\sqrt{5})(\sqrt{2}) = 5 + 2 + 2\sqrt{10} = 7 + 2\sqrt{10}$.
  9. Question 9: Find the value of $(64)^{1/2}$.
    Solution: $(64)^{1/2} = (8^2)^{1/2} = 8^{2 \times \frac{1}{2}} = 8^1 = 8$.
  10. Question 10: Write the rationalizing factor of $\frac{1}{\sqrt{2}}$.
    Solution: The rationalizing factor is $\sqrt{2}$ (since $\sqrt{2} \times \sqrt{2} = 2$, which is rational).
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