Transcendental Numbers: e, π, and What They Aren't

There are two kinds of numbers in elementary mathematics: rational and irrational. Rational numbers are ratios of integers. Irrational numbers — like $\sqrt{2}$ and $\pi$ — are not. The Pythagoreans found this distinction so disturbing that one tradition has them executing the disciple Hippasus for proving $\sqrt{2}$ irrational.

But there’s a finer classification. Among the irrationals, some can be expressed as roots of polynomial equations with integer coefficients — like $\sqrt{2}$, which satisfies $x^2 - 2 = 0$. These are called algebraic numbers. Numbers that can’t be expressed this way — that don’t satisfy any such polynomial — are called transcendental.

Until 1844, no transcendental number was known to exist. By the late 19th century, both $e$ and $\pi$ had been proven transcendental, the Greek problem of squaring the circle was settled (impossible), and the entire theory of transcendence had been opened. Yet over a century later, basic combinations like $e + \pi$ remain unproven, and the field is still full of conjectures awaiting proof.

This article is about what transcendental numbers are, what we know about them, and what we don’t.

The basic distinction

A number $\alpha$ is algebraic of degree $n$ if it satisfies a polynomial of the form

$a_n \alpha^n + a_{n-1} \alpha^{n-1} + \cdots + a_1 \alpha + a_0 = 0$

with integer coefficients $a_i$, and $n$ is the smallest such degree. So:

• Every rational number $p/q$ is algebraic of degree 1 (satisfies $qx - p = 0$).
• $\sqrt{2}$ is algebraic of degree 2 (satisfies $x^2 - 2 = 0$).
• $\sqrt[3]{2}$ is algebraic of degree 3.
• $\sqrt{2} + \sqrt{3}$ is algebraic of degree 4 (satisfies $x^4 - 10x^2 + 1 = 0$).
• $i = \sqrt{-1}$ is algebraic of degree 2 (satisfies $x^2 + 1 = 0$).

A number is transcendental if it is not algebraic of any degree — there’s no polynomial with integer coefficients that has it as a root.

The distinction is subtle but absolute. Every real (or complex) number falls cleanly into one of the two categories. There is no in-between.

Cantor’s surprise

In 1874, Georg Cantor proved a remarkable fact: while there are infinitely many algebraic numbers, the set of algebraic numbers is countable — you can put them into a sequence indexed by the natural numbers. The reals are uncountable (see our piece on Cantor’s infinities). So almost every real number is transcendental.

Specifically: the transcendental numbers form a set of full measure in the reals. If you pick a real number “at random” (in any reasonable measure-theoretic sense), with probability 1 it will be transcendental.

This is a deeply counterintuitive fact. Most numbers we work with explicitly — integers, rationals, square roots, polynomial roots — are algebraic. The transcendentals are the vast majority, but they’re hard to write down explicitly. Most of them have no name and no simple description.

The first transcendental number was constructed by Joseph Liouville in 1844, before Cantor’s result. Liouville built numbers like

$L = \sum_{k=1}^{\infty} \frac{1}{10^{k!}} = 0.1100010000000000000000010000\ldots$

with extreme rapid convergence. He showed that any number with such fast rational approximations cannot be algebraic. This produced the first explicit transcendental — but only by constructing one specifically designed to defy algebraic equations.

$e$ is transcendental

In 1873, Charles Hermite proved that $e \approx 2.71828$ is transcendental. The proof is non-trivial but tractable: Hermite assumed $e$ satisfied some polynomial equation and derived a contradiction by examining a carefully chosen integral.

This was the first transcendence proof for a “natural” number — one that arose in mathematics for reasons unrelated to the construction of transcendentals. (See our piece on $e$ for why $e$ is so fundamental.)

The proof established the technique: assume the number is a root of some polynomial $P(x)$, then construct an integral that must be zero by algebra but is nonzero by analysis. The contradiction proves transcendence.

$\pi$ is transcendental

In 1882, Ferdinand von Lindemann extended Hermite’s technique to prove $\pi$ is transcendental. The proof builds on Hermite’s: it shows that if $\alpha$ is a nonzero algebraic number, then $e^\alpha$ is transcendental. Setting $\alpha = i\pi$ would make $e^{i\pi} = -1$ transcendental — but $-1$ is algebraic (satisfies $x + 1 = 0$). Contradiction. Therefore $i\pi$ is transcendental, hence $\pi$ is transcendental.

This proof has a remarkable consequence: squaring the circle is impossible.

The Greek problem of squaring the circle asks: given a circle, construct a square with the same area using only compass and straightedge. The construction would require the number $\sqrt{\pi}$ — and constructible numbers are restricted to a specific class of algebraic numbers (those reachable by repeated quadratic extensions). Since $\pi$ is transcendental, $\sqrt{\pi}$ is also transcendental, and not constructible.

The classical Greek problem, worked on for over 2000 years, was finally settled negatively in 1882 by an obscure German mathematician using techniques from complex analysis. The Greeks couldn’t have hoped to find the answer with their tools. They were trying to do something fundamentally impossible.

What we still don’t know

Despite the deep results on $e$ and $\pi$ individually, basic questions about their combinations remain open.

Is $e + \pi$ irrational? Unknown. No one has proven that it’s irrational, let alone transcendental. (We know it’s transcendental if $e + \pi$ and $e\pi$ are both algebraic, because then $e$ and $\pi$ would satisfy a quadratic equation with algebraic coefficients — contradicting their known transcendence. So at least one of $e + \pi$ and $e\pi$ is transcendental. But which one — or both — remains unknown.)

Is $e\pi$ rational? Unknown.

Is $\pi^\pi$ transcendental? Unknown.

Is the Euler-Mascheroni constant $\gamma \approx 0.577$ irrational? Unknown — even though $\gamma$ has been computed to billions of decimal places, no one has proven it’s irrational.

These are not exotic edge cases. They’re basic questions about constants that appear constantly in mathematics, and the answers have eluded more than a century of attempts.

The Gelfond–Schneider theorem

Perhaps the deepest 20th-century result on transcendence is the Gelfond–Schneider theorem (proved independently by Aleksandr Gelfond and Theodor Schneider in 1934). It says:

If $\alpha$ and $\beta$ are algebraic numbers with $\alpha \neq 0, 1$ and $\beta$ irrational, then $\alpha^\beta$ is transcendental.

Special cases:

• $2^{\sqrt{2}}$ is transcendental.
• $e^\pi$ is transcendental (because $e^\pi = (e^{i\pi})^{-i} = (-1)^{-i}$, and applying the theorem with $\alpha = -1, \beta = -i$ gives transcendence). This was originally one of Hilbert’s problems.
• $\pi^{\sqrt{2}}$ is transcendental.

The theorem solved Hilbert’s seventh problem from his famous 1900 list. It also illustrates the technique of modern transcendence theory: combine algebraic and analytic information about a number to derive contradictions if the number were algebraic.

Schanuel’s conjecture

The most far-reaching open conjecture in transcendence theory is Schanuel’s conjecture, proposed by Stephen Schanuel in the 1960s:

If $z_1, \ldots, z_n$ are complex numbers that are linearly independent over the rationals, then the field extension $\mathbb{Q}(z_1, \ldots, z_n, e^{z_1}, \ldots, e^{z_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$.

This is technical, but the consequences are enormous. Schanuel’s conjecture would imply:

• $e + \pi$ is transcendental
• $e\pi$ is transcendental
• $e^e$ is transcendental
• $\pi^e$ is transcendental
• All sorts of other combinations are transcendental

It would essentially settle most outstanding transcendence questions about $e$, $\pi$, and exponentials of algebraic numbers. Every transcendence theorem proved so far is a special case of Schanuel.

But Schanuel’s conjecture is no closer to a proof now than it was 60 years ago. It remains a benchmark for what mathematics aspires to but cannot yet reach.

Why this is hard

The transcendence problem has a structural reason for being hard. Algebraic numbers are countable; given a candidate, you can in principle test whether it satisfies any polynomial. But there are uncountably many possible polynomials, infinitely many candidate equations.

The standard proof technique — assume $\alpha$ is algebraic, derive a contradiction — requires constructing some quantity that’s both:

1. Provably nonzero (by some analytic argument)
2. Forced to be zero (by the assumed algebraic relation)

This kind of “contradicting analysis with algebra” is delicate. Most attempts fail because the analytic side and algebraic side don’t match in the right way. Successful proofs (Hermite, Lindemann, Gelfond, Schneider, Baker) each found a clever way to make the analysis and algebra cooperate.

For specific numbers, the proofs are technical and ad hoc. For broader classes (like Schanuel’s conjecture covers), no general proof exists. Transcendence theory is an island of mathematics surrounded by deep waters.

What transcendental numbers teach

The deepest lesson of transcendence theory is that not every interesting number can be reached algebraically. Polynomials with integer coefficients are a powerful tool, but they reach only countably many numbers. The vast majority of real numbers — and many of the most important specific constants in mathematics — lie outside this reach.

This is, in some sense, the irrationality of $\sqrt{2}$ taken to its extreme. The Pythagoreans found that not every length can be expressed as a ratio of integers. Twentieth-century mathematicians found that not every length can be expressed even by polynomial equations in integers. There are layers and layers of “unreachability” in the structure of the real numbers, and we’ve only begun to map them.

For the working mathematician, transcendence theory is mostly an exercise in patience. You try to prove some number is transcendental; the proof is typically heroic and uses unusual techniques; the result is celebrated but doesn’t immediately generalize. Other transcendence questions remain stubbornly open. Progress is made theorem by theorem, century by century.

The transcendentals are everywhere — they’re almost all the real numbers, after all. We just can’t easily say which ones any given real is. That subtle gap, between knowing a class is huge and being able to identify members of it, is one of the more interesting tensions in modern mathematics. It shows up in number theory, in computability theory, in measure theory. The transcendentals are the cleanest example.

The next time you see $\pi$, remember: by Lindemann’s theorem, that one number defeats two thousand years of Greek geometry, settles a problem the world’s greatest mathematicians worked on for generations, and is still surrounded by basic open questions about its arithmetic. Three small letters; an entire branch of mathematics still being explored. Not every constant has that kind of mathematical depth, but $\pi$ does — and so does $e$. They are, in a quite precise mathematical sense, beyond the reach of polynomials.

Frequently asked

Are most numbers transcendental?

Yes — overwhelmingly. The set of algebraic numbers is countable (you can list them, in principle). The reals are uncountable. So almost every real number is transcendental. But specific transcendental numbers are very hard to identify; we've only proven a few are transcendental despite knowing the class is enormous.

Is the proof that π is transcendental complicated?

Yes. Lindemann's 1882 proof is technical and uses complex analysis. The result implies that you cannot 'square the circle' with compass and straightedge — a classical problem the Greeks worked on for over 2000 years. The transcendence of π is the precise mathematical reason that problem is unsolvable.

Are e and π related transcendentally?

We don't know. It's not even known whether e + π is irrational, let alone transcendental. The transcendence of e^π was proven by Gelfond in 1934. But basic combinations — e + π, eπ, e^e — remain mathematical mysteries. This is one of the open questions of analytic number theory.

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