N = set of natural numbers {1,2,3,4,…}

W = set of whole numbers {0,1,2,3,4,…}

Z or I = set of integers {-2,-1,0,1,2,3,…}

Q OR ‘r’ = set of rational number {2/1, 3/5,…}

Q’ or R-Q or ‘s’= set of irrational number {sqrt2, sqrt5,…}

R = set of real numbers

Rational numbers Irrational numbers

The nos. which can be written in the form of p/q, where p & q are integrs and q = 0.

Decimal expansion :- either terminating or non-terminating recurring.

The nos. Which cannot be written in the form of p/q, where p & q are integers and q= 0.

Decimal expansion :- non-terminating non-recurring.

It is true for all rationals of the form p/q (q=0). On division of p by q, two main things happen – either the remainder becomes zero and we get a repeating string of remainders.

CASE I - the remainder becomes zero :-

ex- 1/2 = 0.5 , 639/250 = 2.556

CASE II – the remainder never becomes zero:-

ex- 1/3 = 0.3333…. , 1/7 = 0.142857142857….

1. The sum of difference of a rational number and an irrational number is irrational.

2. The product or quotient of a non-zero rational number with an irrational number is irrational.

3. If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.

STEPS :-

(1) 6.28585 lies between 6 and 7. We divide the number line between 6 and 7 into 10 equal parts andmagnify the distance between them.

(2) Now, 6.28585 lies between 6.2 and 6.3. To get the more accurate visualization, we divide the numberline between them into 10 equal parts and magnify it.

(3) Now, 6.28585 lies between 6.28 and 6.29. We again divide the number line between them into 10equal parts and magnify it.

(4) Now, 6.28585 lies between 6.285 and 6.286. We, thus, divide the number line between them into 10equal parts and magnify it.

(5) Now, 6.28585 lies between 6.2858 and 6.2859. We finally divide the distance between them into 10equal parts and magnify it. We can now mark 6.28585 on the number line.

By following these steps, we will obtain the following set of figures.

Locating root 3 on the number line:-

1. Construct BD of unit length perpendicular to OB .

2. Then using pythagoras theorem we see that

.

3. Using a compass, with centre O and radius OD, draw an arc which intersects the number line at the Q.

4. Then Q corresponds to .

Solution :-

(1) Draw a line and mark a point A on it. Mark points B and C such that AB = 6.7 units and BC = 1 unit.

(2) Find the mid-point of AC and mark it as M. Taking M as the centre and MA as the radius, draw a semi-circle.

(3) From B, draw a perpendicular to AC. Let it meet the semi-circle at D. Taking B as the centre and BD as the radius, draw an arc that intersects the line at E.

Steps :-

Solution :-

Now, the distance BE on this line is units.

1.

2.

3.

4.

5.

6.

or

or

Example 1:Rationalize the denominator of

Solution:We know that , which is a rational number.Now, we have to multiply and divide by .

Example 2:Rationalize the denominator of

Solution:To rationalize the denominator, multiply and divide the number by the conjugate

The conjugate of is .