Representation of Irrational Numbers on the Number Line:

Let us see how we can locate some of the irrational numbers on the number line.

1) Locate √2 on the number line.

Remember that √2 is the length of the diagonal of the square whose side is 1 unit.

• On the number line, point A shows the number 1. Draw line `l` perpendicular to the number line through point A.
  Take point P on line `l` such that OA = AP = 1 unit. 
• Draw seg OP. The ΔOAP formed is a right-angled triangle.
  By Pythagoras theorem, 
  OP² = OA² + AP² = 1² + 1² = 1 + 1 = 2.
  ∴ OP = √2                ....(taking square roots on both sides)
• Now, draw an arc with center O and radius OP. Name the point as Q where the arc intersects the number line. Obviously, distance OQ is `sqrt2`. 
  That is, the number shown by point Q is `sqrt2`. 
• If we mark point R on the number line to the left of O, at the same distance as OQ, then it will indicate the number `- sqrt(2)`.