Rational and Irrational Numbers

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Real Numbers
The union of the set of rational numbers and irrational numbers forms the set of real numbers.
 
Q = {rational numbers}
 
= {irrational numbers}
 
Then = R = {real numbers}
 
To represent a rational number on the number line
 
Mark the following rationals on the number line:
 
 
 
To represent an irrational number on the number line
 
To mark , an irrational number, on the number line.
 
Let OA = AB = 1 unit and
 
Then in rt.
 
 
= 1 + 1
 
 
 
 
Cut off OP = OB = on the number line.
 
Then, the point P represents on the number line.
 
 
Using Pythagoras' theorem in rt.
 
 
OB = , if AO = 2 and AB = 1
 
Cut off OP = OB =
 
Then the point P represents on the number line.
 
In the same way can be represented on the number line.
 
 
(i) For every real number, there is a corresponding point on the number line.
 
(ii) For every point on the number line, there exists a real number.
 
 
Express with rational denominator.
 
 
 
 
Express with rational denominator.
 
 
 
 
  
     
   
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