Number Theory/Irrational Rational and Transcendental Numbers

Definitions

Rational numbers are numbers which can be expressed as a ratio of two integers.

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 However, a decimal which does not repeat after a finite number of decimals is NOT a rational number. Examples include
$1.25,-0.333333,999\dots$

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 $ℝ$.

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

Transcendental numbers are irrational numbers which are not the root of any polynomial equation.

For example, $\sqrt{2}$ is a root of the polynomial equation $x^2-2=0$ and so isn't transcendental (but it is irrational).

However, it is known that $\pi ,e$ are not the roots of any possible polynomial and so they are transcendental.

The set of transcendental numbers is infinite.