Rational, irrational, transcendental

What really needs to be proved

A real number $x$ is rational if it can be written as a quotient of two integers:

$$x \in \mathbb{Q} \iff \exists (p, q) \in \mathbb{Z} \times \mathbb{N}^*,\quad x = \frac{p}{q}$$

It is irrational otherwise. Proving that a number is irrational therefore means proving a negative property: no fraction, however complicated, will do. That is why almost all such proofs proceed by contradiction — we assume the fraction is given, and derive an arithmetic contradiction from it.

Numerical computation proves nothing

The first decimal digits of $e$ betray no structure:

$$e = 2.718\,281\,828\,459\,045\,235\,360\ldots$$

The pattern $1828$ appears twice in a row, which is pure coincidence. Above all, no finite computation can settle the matter: every rational number with a very large denominator has a periodic decimal expansion, but its period may be gigantic.

Number	Decimal expansion	Nature
$\dfrac{1}{3}$	$0.333\ldots$, period $1$	rational
$\dfrac{1}{7}$	$0.\overline{142857}$, period $6$	rational
$\dfrac{1}{97}$	period of $96$ digits	rational
$\sqrt{2}$	infinite, non-periodic	algebraic irrational
$e$	infinite, non-periodic	transcendental irrational

Irrational is not transcendental

Two notions not to be confused.

A real number is algebraic if it is a root of a nonzero polynomial with integer coefficients, and transcendental otherwise. Thus $\sqrt{2}$ is irrational, but algebraic, since it is a root of $X^2 - 2$. The number $e$, on the other hand, is irrational and transcendental: Hermite proved it in 1873, and Lindemann did the same for $\pi$ in 1882.

$$\mathbb{Q} \subset \left\{\text{algebraic numbers}\right\} \subset \mathbb{R}$$

The irrationality of $e$ is therefore a strictly weaker result than its transcendence — but its proof fits in half a page, whereas Hermite's requires several pages of analysis.

Classic pitfall

Writing "$e$ is irrational because its decimal expansion is infinite". That argument is wrong: $\dfrac{1}{3}$ also has an infinite decimal expansion. The correct criterion is the absence of periodicity, and it is useless in practice, since one can never verify the absence of a period from a finite number of decimal digits.