Proof of the transcendence of pi from the Lindemann–Weierstrass theorem

Builds on Transcendental element Algebraic number The algebraic numbers form a field Lindemann–Weierstrass theorem Euler's formula

Proof

We argue by contradiction.

Suppose that $\pi$ is not transcendental over $\Q$ . Then $\pi$ is an algebraic number.
The number $i$ is algebraic, since it is a root of $x^2 + 1$ .
The algebraic numbers form a field, so $\alpha_1 = i\pi$ is algebraic. Also $\alpha_2 = 0$ is algebraic, and $\alpha_1 \neq \alpha_2$ because $\pi \neq 0$ .
By Euler's formula, $e^{i\pi} = \cos \pi + i \sin \pi = -1$ . Also $e^0 = 1$ , from the series of the exponential.
With $\beta_1 = \beta_2 = 1$ , which are algebraic, we get $\beta_1 e^{\alpha_1} + \beta_2 e^{\alpha_2} = -1 + 1 = 0$ .
By the Lindemann–Weierstrass theorem, this forces $\beta_1 = \beta_2 = 0$ . This contradicts $\beta_1 = 1$ .
So $\pi$ is transcendental over $\Q$ .

Idea

Euler's identity $e^{i\pi} + 1 = 0$ is a linear relation, with algebraic coefficients, between exponentials of $i\pi$ and $0$ . The Lindemann–Weierstrass theorem forbids such a relation when the exponents are distinct algebraic numbers. So $i\pi$ , and therefore $\pi$ , cannot be algebraic.

Reading path

To understand this page from the ground up, read these 205 pages in order. The foundations come first. The path takes the proofs Proof that the algebraic numbers form a field, Proof of the limit of a sum of sequences, Proof of cancellation in an integral domain, Proof of the commutativity of addition by induction, Proof of the universal property of the polynomial ring, Proof of the tower law, Proof that R[x] is an integral domain iff R is, Proof that finite extensions are algebraic, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof that the degree of a simple extension is the degree of the minimal polynomial, Proof of the first isomorphism theorem for rings, Proof that a ring homomorphism is injective iff its kernel is zero, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof of the correspondence theorem for rings, Proof that the absolute value is multiplicative, Proof of the existence of square roots of nonnegative reals, Proof that i squared is minus one, Proof that R/M is a field iff M is maximal, Proof that the only ideals of a field are zero and the whole field, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial, Proof of Zorn's lemma, Proof of the well-ordering principle from strong induction, Proof that the minimal polynomial is irreducible, Proof that the minimal polynomial divides every polynomial vanishing at the element, Proof that convergent sequences are bounded, Proof that K[x]/(f) is a field iff f is irreducible, Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal, Proof of Euler's formula by power series, Proof that every vector space has a basis, by Zorn's lemma, Proof that Euclidean domains are principal ideal domains, Proof of the universal property of the quotient ring, Proof that K[x] is a Euclidean domain.

Algebraic number definition
Complex numbers object
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Group concept
Imaginary unit object
Integers object
Linear combination concept
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Spanning set concept
Vector space definition
Transcendental number definition
Complex exponential definition
Power series in the complex plane definition
Real and imaginary parts definition
Proof that i squared is minus one proof
Set definition
Ring definition
Ring of integers statement
Successor function definition
Disjunction definition
Implication definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Supremum definition
Linear combination definition
Linear subspace definition
Lindemann–Weierstrass theorem statement
i squared is minus one statement
Family of sets definition
Ordered pair definition
Ideal definition
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Subring definition
Union definition
Unit of a ring definition
Zero divisor definition
Addition of natural numbers definition
The successor function is injective statement
Zero is not a successor statement
Existential quantifier definition
The rationals form a field statement
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Linear independence definition
Completeness axiom statement
Linear span definition
Cartesian product definition
Maximal ideal definition
Two-sided ideal definition
Ideal generated by a subset definition
Divisibility in a commutative ring definition
Degree of a polynomial definition
Polynomial evaluation definition
Proof of the universal property of the polynomial ring proof
Kernel of a ring homomorphism definition
Ring isomorphism definition
Field definition
Integral domain definition
Union of a family of sets definition
Subset definition
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Convergent sequence definition
Spanning set definition
Principal ideal definition
Quotient ring definition
Leading coefficient definition
Root of a polynomial definition
Binary relation definition
Evaluation homomorphism definition
Proof that the only ideals of a field are zero and the whole field proof
Proof of cancellation in an integral domain proof
Order on the reals definition
Subfield definition
Irreducible element definition
Euclidean domain definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Finite-dimensional vector space definition
Basis definition
Monic polynomial definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Proof that R/M is a field iff M is maximal proof
Proof of the universal property of the quotient ring proof
Principal ideal domain definition
Proof of the degree of a product over an integral domain proof
Universal property of the polynomial ring statement
The only ideals of a field are zero and the whole field statement
Irreducible polynomial definition
Cancellation in an integral domain statement
Field extension definition
Function definition
Strong induction statement
Dimension definition
Dimension theorem for vector spaces statement
Preorder definition
Division algorithm for polynomials statement
Universal property of the quotient ring statement
Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal proof
Degree of a product statement
Algebraic element definition
Intermediate field definition
Proof of the well-ordering principle from strong induction proof
Degree of a field extension definition
Choice function definition
Codomain definition
Image of a set definition
Injective function definition
Preimage of a set definition
Partial order definition
Minimal polynomial of an algebraic element definition
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Transcendental element definition
Proof that R[x] is an integral domain iff R is proof
Finite extension definition
Algebraic extension definition
Field generated by adjoining elements definition
Proof that a ring homomorphism is injective iff its kernel is zero proof
Well-ordering principle statement
Axiom of choice statement
Surjective function definition
Maximal element definition
Total order definition
Upper bound definition
Proof that the minimal polynomial divides every polynomial vanishing at the element proof
Proof that Euclidean domains are principal ideal domains proof
R[x] is an integral domain iff R is statement
Proof that the minimal polynomial is irreducible proof
Simple extension definition
A ring homomorphism is injective iff its kernel is zero statement
Bijective function definition
Chain definition
The minimal polynomial divides every polynomial vanishing at the element statement
Proof of the first isomorphism theorem for rings proof
Proof that K[x] is a Euclidean domain proof
The minimal polynomial is irreducible statement
Proof of the correspondence theorem for rings proof
Euclidean domains are principal ideal domains statement
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Equinumerous sets definition
Correspondence theorem for rings statement
First isomorphism theorem for rings statement
K[x] is a Euclidean domain statement
Zorn's lemma statement
Cardinality definition
The reals form an ordered field statement
R/M is a field iff M is maximal statement
Absolute value definition
Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
Proof that every vector space has a basis, by Zorn's lemma proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that K[x]/(f) is a field iff f is irreducible proof
Every vector space has a basis statement
Proof of the epsilon-N definition of the limit of a sequence proof
Every linearly independent set extends to a basis statement
Existence of square roots of nonnegative reals statement
Triangle inequality statement
Absolute value is multiplicative statement
K[x]/(f) is a field iff f is irreducible statement
Square root definition
Proof that finite extensions are algebraic proof
Proof of the tower law proof
Epsilon-N definition of the limit of a sequence statement
Modulus of a complex number definition
Proof of the limit of a sum of sequences proof
Finite extensions are algebraic statement
Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial proof
Tower law statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof that convergent sequences are bounded proof
K(a) is isomorphic to K[x] modulo the minimal polynomial statement
Limit of a sum of sequences statement
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are bounded statement
Proof of the limit of a product of sequences proof
Proof that the degree of a simple extension is the degree of the minimal polynomial proof
Degree of a simple extension is the degree of the minimal polynomial statement
Limit of a product of sequences statement
Proof of Euler's formula by power series proof
Proof that the algebraic numbers form a field proof
Euler's formula statement
The algebraic numbers form a field statement
Proof of the transcendence of pi from the Lindemann–Weierstrass theorem proof

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4 pages build on this one.

Tag	`26XC0EM9M6`
URI	`m2://26XC0EM9M6`
Kind	proof
Writing	W0 stub
MSC	11J81
Proves	Transcendence of pi
Uses	Transcendental element, Algebraic number, The algebraic numbers form a field, Lindemann–Weierstrass theorem, Euler's formula
Changed	4 November 2026

Proof of the transcendence of pi from the Lindemann–Weierstrass theorem

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