Revisiting Rational and Irrational Numbers

Rational number:
A number which can be written in the form $\frac{\text{a}}{\text{b}}$ 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 $\frac{\text{a}}{\text{b}}$, 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 $\frac{\text{p}}{\text{q}}$ 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 $\frac{\text{p}}{\text{q}}$ terminates.

Theorem:
If a rational number is a terminating decimal, it can be written in the form $\frac{\text{p}}{\text{q}}$ , 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 $\frac{\text{p}}{\text{q}}$ 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 $\frac{\text{p}}{\text{q}}$ does not terminate and is recurring.