Number Theory/Irrational Rational and Transcendental Numbers

Definitions
Rational numbers are numbers which can be expressed as a ratio of two integers (with a non-null denominator).

This includes fractional representations such as $$\frac{3}{4} \,, -\frac{27}{3} \, $$ etc.

A rational number can also be expressed as a termininating or recurring decimal. Examples include $$1.25, -0.333333 , 0.999 \ldots$$

However, a decimal which does not repeat after a finite number of decimals is NOT a rational number.

One other representation that is sometimes used is that of a ratio e.g. $$5:4 \, $$

The entire (infinite) set of rational numbers is normally referenced by the symbol $$\mathbb{Q} \, $$.

Irrational numbers are all the rest of the numbers - such as $$\sqrt{2}, \pi , e \, $$

Taken together, irrational numbers and rational numbers constitute the real numbers - designated as $$\mathbb{R} \, $$.

The set of irrational numbers is infinite - indeed there are "more" irrationals than rationals (when "more" is defined precisely).

Algebraic numbers are numbers which are the root of some polynomial equation with rational coefficients. For example, $$\sqrt{2} $$ is a root of the polynomial equation $$x^2 - 2 = 0 \, $$ and so it is an algebraic number (but irrational).

Transcendental numbers are irrational numbers which are not the root of any polynomial equation with rational coefficients. For example, $$\pi, e \, $$ are not the roots of any possible polynomial and so they are transcendental.

The set of transcendental numbers is infinite - indeed there are "more" transcendental than algebraic numbers (when "more" is defined precisely).