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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.
 


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 – OP

OQ = 4. 2 – 2.6 = 1.6
 


By Pythagoras theorem,

QS2 = OS2 – OQ2

= (2.6) 2 - (1.6) 2

= 6.76 - 2.56

 

QS2 = 4.2

Þ QS =

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

 

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:

 

Example :

(a) 2 + 3 = 5

(b) (5 ) (8)

(c) (2 + ) ( 2 - ) = 4 – 7= - 3

(d) 





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