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.