# Operations on Real Numbers

Let us see how to represent the irrational number on a number line using geometrical method.

Let us take x = 4.2, find geometrically.

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Draw a line PQ = 4.2 and then extend the line PQ to R for 1 unit.

Mid-point of PR =

Draw a semicircle, such that taking OP as radius and OR as radius.

Now, draw a line QS ^ PR.

OR= OS = OP = 2.6 = radius

OQ =PQ â€“ OP

OQ = 4. 2 â€“ 2.6 = 1.6

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By Pythagoras theorem,

QS^{2} = OS^{2} â€“ OQ^{2}

= (2.6)^{ 2} - (1.6)^{ 2}

= 6.76 - 2.56

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QS^{2 }= 4.2

Ãž QS =

Before extending the idea of square roots, let us recall their identities.

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Consider * p* and *q* are positive rational numbers, then:

If and are rational numbers, and if the product is also a rational number, then we can say that and are rationalizing factor of each other.

Let us see some examples applying the above identities:

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**Example :**

(a) 2 + 3 = 5

(b) (5 ) (8)

(c) (2 + ) ( 2 - ) = 4 â€“ 7= - 3

(d)Â