Chapter 1: Number Systems

Exercise 1.2 - Detailed Solutions

Exercise 1.2 - Step-by-Step Solutions

❓Question 1: True or False

(i) Every irrational number is a real number.

Step 1: Define sets

  • Real numbers = Rational ∪ Irrational
  • So irrationals are PART OF real numbers

Answer: TRUE ✓

(ii) Every point on number line is of form √m

Step 1: Think of negative numbers

  • -3 is on the number line
  • Can we write -3 = √m for any natural m?
  • √m is always positive for natural m!

Answer: FALSE ✗ (Negative numbers are on number line but arent √m)

(iii) Every real number is irrational

Step 1: Find a counterexample

  • Is 5 real? Yes!
  • Is 5 irrational? No, 5 = 5/1 is rational!

Answer: FALSE ✗

❓Question 2

Are all square roots irrational?

✅Solution:

Step 1: Consider perfect squares

  • √1 = 1 (rational!)
  • √4 = 2 (rational!)
  • √9 = 3 (rational!)
  • √16 = 4 (rational!)

Step 2: Non-perfect squares

  • √2 = irrational
  • √3 = irrational
  • √5 = irrational

Answer: NO! Square roots of perfect squares are rational. Only square roots of non-perfect squares are irrational.

❓Question 3

Show √5 on number line

✅Solution:

Step 1: Use Pythagoras We need √5 = √(4 + 1) = √(2² + 1²)

Step 2: Construction

  1. Draw line OA = 2 units on number line
  2. At A, draw perpendicular AB = 1 unit
  3. Join OB
  4. By Pythagoras: OB = √(4+1) = √5

Step 3: Transfer to number line 5. With O as center, radius OB, draw arc 6. Arc meets number line at P 7. OP = √5 ✓

❓Question 4: Square Root Spiral

This is the Theodorus Spiral!

Starting from a right triangle with sides 1 and 1:

  • Hypotenuse = √2
  • Add perpendicular of 1, new hypotenuse = √3
  • Continue to get √4, √5, √6...

This creates a beautiful spiral showing all square roots!

Visualizer
Square Root Spiral (Theodorus)
Showing √2 to √1

Interactive Visualization