**Representing Irrational Numbers On The Number Line**

**Represent **** √2 & √3**** on the number line:**

Greeks discovered this method. Consider a unit square OABC, with each side 1 unit in lenght. Then by using pythagoras theorem

\(OB=\sqrt{1+1}=\sqrt{2}\)

Now, transfer this square onto the number line making sure that the vertex O coincides with zero

With O as centre & OB as radius, draw an arc, meeting OX at P. Then

OB = OP = √2 units

Then, the point represents √2 on the number line

Now draw, BD ⊥ OB such that BD = 1 unit join OD. Then

OD = \(\sqrt{{{(\sqrt{2})}^{2}}+{{(1)}^{2}}}=\sqrt{3}\) = units With O as centre & OC as radius, draw an arc, meeting OX at Q. Then

OQ = OD = √3 units

Then, the point Q represents √3 on the real line

**Remark:** In the same way, we can locate √n for any positive integer n, after \(\sqrt{n-1}\) has been located.

**Existence of √n for a positive real number:**

The value of √4.3 geometrically : –

Draw a line segment AB = 4.3 units and extend it to C such that BC = 1 unit.

Find the midpoint O of AC.

With O as centre and OA a radius, draw a semicircle.

Now, draw BD ⊥ AC, intersecting the semicircle at D. Then, BD = √4.3 units.

With B as centre and BD as radius, draw an arc, meeting AC produced at E.

Then, BE = BD = √4.3 units