Rational number:
A number which can be written in the form
where
a and b are integers and
b ≠ 0 is called a rational number.
Rational numbers are of two types depending on whether their decimal form is terminating or recurring.
Irrational number:
A number which cannot be written in the form , where a and b are integers and b ≠ 0 is called a irrational number. Irrational numbers which have non-terminating and non-repeating decimal representation.
The sum or difference of a two irrational numbers is also rational or an irrational number.
The sum or difference of a rational and an irrational number is also an irrational number.
Product of a rational and an irrational number is also an irrational number.
Product of a two irrational numbers is also rational or an irrational number.
Theorem:
Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
Theorem:
If
is a rational number, such that the prime factorisation of q is of the form 2
a5
b, where a and b are positive integers, then the decimal expansion of the rational number
terminates.
Theorem:
If a rational number is a terminating decimal, it can be written in the form
, where p and q are co prime and the prime factorisation of q is of the form 2
a5
b, where a and b are positive integers.
Theorem:
If
is a rational number such that the prime factorisation of q is not of the form 2
a5
b where a and b are positive integers, then the decimal expansion of the rational number
does not terminate and is recurring.