**Important Concepts and Formulas**

1. A number is called a rational number,
if it can be written in the form p/q , where p, q ∈ Z and q ≠ 0.

2. A number is called an irrational
number, if it cannot be written in the form p/q , where p,
q ∈ Z 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
rational.

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 irrational.

5. The collection of rational and
irrational numbers is called the collection of real numbers.

6. There is a unique real number
corresponding to every point on the number line. Also, corresponding to each
real number, there is a unique point on the number line.

7. If r is a rational number and s is an
irrational number, then r + s, r – s, r × s and r/s are irrational numbers, r ≠ 0.

8. For positive real numbers a and b, the following identities
hold true:

9. To rationalize the denominator of 1/(√a + b), we multiply this by (√a – b) / (√a – b), where a and b are
integers.

10. Let a ˃ 0 be a real number and p and q be
rational numbers, then

i.

*a*^{p}.a^{q}= a^{p + q}^{ }ii.*(a*^{p})^{q}= a^{pq}
iii.

*a*iv.^{p}/a^{q}= a^{p – q}*a*^{p}b^{p}= (ab)^{p}