# 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 ${\displaystyle {\frac {3}{4}}\,,-{\frac {27}{3}}\,}$  etc.

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

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. ${\displaystyle 5:4\,}$

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

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

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

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, ${\displaystyle {\sqrt {2}}}$  is a root of the polynomial equation ${\displaystyle x^{2}-2=0\,}$  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, ${\displaystyle \pi ,e\,}$  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).