Rational and Irrational Numbers

Rational numbers are the numbers that can be represented as the quotient of two integers p and q, where q is not equal to zero. If the set of rational numbers is denoted by Q , then

Q = { all x, where x = p / q , p and q are integers, q is not zero}

Rational numbers can be represented as:

(1) Integers: (4 / 2) = 2, (12 / 4) = 3
(2) Fractions: 3 / 4, 13 / 3
(3) Terminating Decimals: (3 / 4) = 0.75, (6 / 5) = 1.2
(4) Repeating Decimals: (13 / 3) = 4.333....., (4 / 11) = .363636......

Conversely, irrational numbers are the numbers that cannot be represented as the quotient of two integers, i.e., irrational numbers cannot be rational numbers and vice-versa. If the set of irrational numbers is denoted by H, then

H = { all x, where there exists no integers p and q such that x = p / q, q is
not zero }

Typical examples of irrational numbers are the numbers p and e, as well as the principal roots of rational numbers. They can be expressed as non-repeating decimals, i.e., the numbers after the decimal point do not repeat their pattern.