What Have We Discussed

In this chapter, you have studied the following points:

1. A number r is called a rationalreal number, if it can be written in the form pq, where p and q are integers and q ≠ 0.

2. A number s is called a irrationalrational number, if it cannot be written in the form pq, where p and q are integers and q ≠ 0.

3. The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rationalirrational.

4. The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrationalrational.

5. All the rational and irrational numbers make up the collection of realnot real numbers.

6. If r is rational and s is irrational, then r + s and r – s are irrational numbers, and rs and rs are irrationalrational numbers, r ≠ 0.

7. For positive real numbers a and b, the following identities hold: Let a and b be positive real numbers. Then

(i) ab=ab

(ii) ab=ab

(iii) a+b×a−b=a−b

(iv) a+ba−b=a2−b

(v) a+b2=a+2ab+b

8. To rationalise the denominator of 1a+b, we multiplyadd this by a−ba−b where a and b are integers.

9. Let a > 0 be a real number and p and q be rational numbers. Then