What We Have Discussed

A number which can be written in the form pq, where p and q are integers and q ≠ 0 is called a rational number.

A number which cannot be written in the form pq, for any integers p, q and q ≠ 0 is called an irrational number.

The decimal expansion of a rational number is either terminating or non-terminating .

The decimal expansion of an irrational number is non-terminating and .

The collection of all rational and irrational numbers is called Real numbers.

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

If q is rational and s is irrational, then q+s, q-s, qs and q/s are numbers.

If n is a natural number other than a perfect square, then n is an number.

The following identities hold for positive real numbers a and b:

(i) ab = ab

(ii) ab = ab (b ≠ 0)

(iii) a+ba−b = -

(iv) a+ba−b = a2 -

(v) a+b2 = + 2ab +

(vi) a + b + 2ab = a + b

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

Let a > 0, b > 0 be a real number and p and q be rational numbers. Then:

(i) apag = ap+g

(ii) apg = apg

(iii) apag = ap−g

(iv) apbp = abp

If 'n' is a positive integer > 1 and 'a' is a positive rational number but not nth power of any rational number then ᶯ√a or aⁿ is called a surd of nth order.