Show how √3 can be represented on the number line.

1. Let line AB be of 1 unit on a number line.
2. At B, draw a perpendicular line BC of length 1 unit.
3. Join CA.
4. Now, ABC is a right-angled triangle. Applying Pythagoras' theorem:
   • AB² + BC² = CA²
   • 1² + 1² = CA²
   • CA² = 2
   • ⇒ CA = √2
   Thus, CA is a line of length √2 units.
5. Taking CA as the base, draw a perpendicular line from point A of length 1 unit. Label the endpoint of this line as D.
6. Join CD.
7. Now, ADC is a right-angled triangle. Applying Pythagoras' theorem again:
   • CA² + AD² = CD²
   • (√2)² + 1² = CD²
   • 2 + 1 = CD²
   • CD² = 3
   • ⇒ CD = √3
   Thus, CD is a line of length √3 units.
8. Taking CD as a radius and D as the center, draw an arc touching the number line. The point at which the number line is intersected by the arc is at √3 distance from 0 because it is the radius of the circle whose center is D.

Thus, √3 is represented on the number line as shown in the figure.