Transcendental numbers — π and e

What just happened?

Real numbers split into two classes. An algebraic number is one that is a root of some nonzero polynomial with rational (integer) coefficients. This class includes all rational numbers — e.g. $\tfrac12$ is a root of the equation $2x - 1 = 0$ — as well as some irrational numbers, such as $\sqrt2$, $\sqrt[5]{7}$ or the golden ratio $\varphi$. A transcendental number is one that is NOT a root of any such polynomial — it “transcends” algebra, hence the name.

√2 is algebraic — it is a root of a rational polynomial:

Although $\sqrt2$ is irrational, it “closes up” under an equation — which is why it counts as an algebraic number. Likewise $\tfrac12$ satisfies $2x - 1 = 0$.

π and e are transcendental — no rational polynomial hits them:

Proving that a specific number is transcendental is very hard. Charles Hermite was the first to show the transcendence of $e$ (1873), and Ferdinand von Lindemann the transcendence of $\pi$ (1882) — which incidentally settled the ancient problem of squaring the circle: it is impossible with compass and straightedge precisely because $\pi$ is transcendental.

A surprising proportion:

Georg Cantor showed that there are only countably many algebraic numbers (cardinality $\aleph_0$), but uncountably many real numbers (cardinality continuum $\mathfrak{c} > \aleph_0$). Since the algebraic ones are “few” and the reals “incomparably more numerous”, almost every real number is transcendental — even though we know surprisingly few explicit examples.

Why this matters for your exams:

Distinguishing rational, irrational-algebraic and transcendental numbers organises the world of real numbers and recurs in problems about irrationality proofs, about roots of polynomials, and about what cannot be constructed with compass and straightedge (such as squaring the circle).

Remember: a number is algebraic when it is a root of some polynomial with rational coefficients (like $\sqrt2$ for $x^2-2$), and transcendental when it is a root of none (like $e$ and $\pi$). Transcendentals are incomparably more numerous.