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. ap.aq
= ap + q ii. (ap)q = apq
iii. ap/aq
= ap – q iv. apbp = (ab)p