Number Systems

NCERT Solutions • Class 9 Maths • Chapter 1
Exercise 1.2
1. State whether the following statements are true or false. Justify your answers.
Solution:
  1. (i) Every irrational number is a real number.
    True.
    Reason: The collection of Real Numbers ($\mathbb{R}$) is made up of all rational and all irrational numbers.
  2. (ii) Every point on the number line is of the form $\sqrt{m}$, where $m$ is a natural number.
    False.
    Reason: Negative numbers (like $-2, -3$) appear on the number line, but the square root of a natural number $\sqrt{m}$ is always positive. No natural number $m$ exists such that $\sqrt{m} = -2$.
  3. (iii) Every real number is an irrational number.
    False.
    Reason: Real numbers contain rational numbers too. For example, $2$ is a real number, but it is rational, not irrational.
2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Solution: No, the square roots of all positive integers are not irrational.

Example: Consider the number 4. $$ \sqrt{4} = 2 $$ Here, 2 is a rational number (since it can be written as $\frac{2}{1}$).

Other examples: $\sqrt{9} = 3$, $\sqrt{16} = 4$, $\sqrt{25} = 5$. All these are rational.
3. Show how $\sqrt{5}$ can be represented on the number line.
Solution: To represent $\sqrt{5}$ on the number line, we use the Pythagorean Theorem.
We know that $5 = 4 + 1 = 2^2 + 1^2$. $$ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 $$ $$ (\sqrt{5})^2 = 2^2 + 1^2 $$
Construction Steps:
  • Draw a number line and mark a point $O$ representing zero (0) and point $A$ representing 2. Thus, $OA = 2$ units.
  • At point $A$, draw a perpendicular $AB$ of length 1 unit. So, $AB = 1$ unit.
  • Join $O$ and $B$. By Pythagoras theorem in right-angled $\triangle OAB$: $$ OB = \sqrt{OA^2 + AB^2} = \sqrt{2^2 + 1^2} = \sqrt{4+1} = \sqrt{5} $$
  • Now, taking $O$ as the center and $OB$ ($\sqrt{5}$) as the radius, draw an arc that cuts the number line at a point $D$.
  • The point $D$ represents $\sqrt{5}$ on the number line.
4. Classroom activity (Constructing the ‘square root spiral’).
Construction Process: The ‘Square Root Spiral’ is constructed by creating a sequence of right-angled triangles.

  • Step 1: Start at a point $O$ and draw a line segment $OP_1$ of unit length (1 unit).
  • Step 2: Draw a line segment $P_1P_2$ perpendicular to $OP_1$ of unit length. Join $OP_2$.
    Result: $OP_2 = \sqrt{1^2 + 1^2} = \sqrt{2}$.
  • Step 3: Draw a line segment $P_2P_3$ perpendicular to $OP_2$ of unit length. Join $OP_3$.
    Result: $OP_3 = \sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{3}$.
  • Step 4: Draw a line segment $P_3P_4$ perpendicular to $OP_3$ of unit length. Join $OP_4$.
    Result: $OP_4 = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{4} = 2$.
  • Conclusion: Repeating this process for $P_n$ generates hypotenuses of lengths $\sqrt{2}, \sqrt{3}, \sqrt{4}, \dots \sqrt{n}$, creating a spiral shape.
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