# Transcendental number
A **transcendental number** is a complex number that is not algebraic, that is, not a root of any non-zero polynomial with integer (equivalently, rational) coefficients[1](https://encyclopediaofmath.org/wiki/Transcendental_number) • [2](https://www.worldscientific.com/worldscibooks/10.1142/q0085). The definition is purely negative, and this is the root cause of the difficulty of the subject: to prove a number transcendental one must rule out every polynomial equation it could satisfy, not just exhibit a property it has[3](https://simonrs.com/eulercircle/irpw2022/varun-transcendental-paper.pdf). The sets nest strictly: the rationals sit inside the algebraic numbers, which sit inside the complex numbers, and the complement of the middle layer is the transcendental one[2](https://www.worldscientific.com/worldscibooks/10.1142/q0085).
| Key fact | Detail |
|---|---|
| Definition | A complex number τ with P(τ) ≠ 0 for every non-zero polynomial P with rational coefficients[2](https://www.worldscientific.com/worldscibooks/10.1142/q0085) |
| First examples | Liouville, 1844, via numbers too well approximated by rationals; the Liouville constant ∑k≥1 10−k! • [1](https://encyclopediaofmath.org/wiki/Transcendental_number) • [2](https://www.worldscientific.com/worldscibooks/10.1142/q0085) |
| Analytic constants | e (Hermite, 1873), π (Lindemann, 1882), eπ and 2√2 (Gelfond and Schneider, 1934)[1](https://encyclopediaofmath.org/wiki/Transcendental_number) • [4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs) |
| Still open | e + π, eπ, log π, πe, Γ(1/5), ζ(3), and the Euler–Mascheroni constant[4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs) |
| Abundance | Almost all real numbers are transcendental (Borel), yet the vast majority of transcendental numbers remain unproven as such[1](https://encyclopediaofmath.org/wiki/Transcendental_number) • [3](https://simonrs.com/eulercircle/irpw2022/varun-transcendental-paper.pdf) |
| Approximation barrier | An algebraic irrational ρ satisfies \|ρ − p/q\| > c(ρ)/qd for degree d; Roth (1955) showed algebraic irrationals have irrationality measure 2, essentially optimally[5](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_Number_Theory_(Veerman)/01%3A_A_Quick_Tour_of_Number_Theory/1.03%3A_Algebraic_and_Transcendental_Numbers) |
| Central conjecture | Schanuel: for Q-linearly independent x1,…,xn, at least n of the 2n numbers xi, exi are algebraically independent[4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs) |
## History
**Liouville's construction, 1844.** Joseph Liouville gave the first existence proof and explicit examples of transcendental numbers, building on his observation that irrational algebraic numbers do not have very good rational approximations[1](https://encyclopediaofmath.org/wiki/Transcendental_number). His first example, now called the Liouville constant, is the decimal with 1s in the factorial positions: ∑k≥1 10−k! = 0.110001000000000000000001…; Liouville showed it is approximated by rationals too well for any algebraic irrational[2](https://www.worldscientific.com/worldscibooks/10.1142/q0085).
**The analytic constants.** Proving transcendence for constants of analysis proved harder. [Charles Hermite](https://www.edgepedia.org/charles-hermite) proved e transcendental in 1873, and in 1882 Carl Lindemann adapted Hermite's method to prove π transcendental, giving the final negative answer to the ancient Greek problem of squaring the circle; an earlier attempt by James Gregory in 1667 had failed because the problem exceeded seventeenth-century methods[4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs) • [6](https://www.britannica.com/topic/algebraic-versus-transcendental-objects-1084407). Lindemann also proved the transcendence of logarithms of algebraic numbers in general, and [Alexander Gelfond](https://www.edgepedia.org/alexander-gelfond) proved that of 2√2 • [1](https://encyclopediaofmath.org/wiki/Transcendental_number).
**Hilbert's seventh problem.** Hilbert's seventh problem, posed in 1900, asked whether αβ is transcendental when α ≠ 0, 1 is algebraic and β is an algebraic irrational. Gelfond and [Theodor Schneider](https://www.edgepedia.org/theodor-schneider) answered it simultaneously and independently in 1934[1](https://encyclopediaofmath.org/wiki/Transcendental_number) • [4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs). [Alan Baker](https://www.edgepedia.org/alan-baker-mathematician)'s work of the 1960s on linear forms in logarithms extended these results and earned him the [Fields Medal](https://www.edgepedia.org/fields-medal)[7](https://arxiv.org/pdf/2603.24823).
## Key theorems and how they work
**Lindemann's theorem** states that the exponential function ez takes transcendental values at every non-zero algebraic point z[8](https://encyclopediaofmath.org/wiki/Lindemann_theorem). Its 1885 extension, the **Lindemann–Weierstrass theorem**, says that for non-zero algebraic coefficients ai and pairwise distinct algebraic αi, the sum a1eα1 + … + ameαm is non-zero; equivalently, if β1,…,βm are algebraic numbers linearly independent over the rationals, then eβ1,…,eβm are linearly independent over the algebraic numbers[8](https://encyclopediaofmath.org/wiki/Lindemann_theorem). From it one deduces that π is transcendental (since eiπ = −1 is algebraic, iπ cannot be algebraic), that squaring the circle has no ruler-and-compass solution, and that sin z, cos z, tan z, and ln z take transcendental values at non-zero algebraic arguments, with ln z for algebraic z ≠ 0, 1[8](https://encyclopediaofmath.org/wiki/Lindemann_theorem). This is how ln 2 joins the proved list.
**Gelfond–Schneider** handles the case Lindemann cannot reach: αβ is transcendental whenever α is algebraic, α ∉ {0, 1}, and β is algebraic irrational[1](https://encyclopediaofmath.org/wiki/Transcendental_number). The theorem yields the transcendence of both 2√2 and eπ • [9](https://mathworld.wolfram.com/TranscendentalNumber.html).
**Baker's theorem** unifies and extends these statements. If log α1,…,log αn are logarithms of algebraic numbers linearly independent over Q, then 1, log α1,…,log αn are linearly independent over the field of algebraic numbers; the Hermite–Lindemann theorem is the case n = 1 and Gelfond–Schneider the two-logarithm case, and Baker's proofs rest on a generalization of Gelfond's method[10](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Strategy.pdf). Crucially, one can give effective lower bounds for such linear forms in logarithms, which has profound consequences for the theory of Diophantine equations[1](https://encyclopediaofmath.org/wiki/Transcendental_number). Baker's 1975 monograph systematically covers this theory along with Schmidt's generalization of the [Thue–Siegel–Roth theorem](https://www.edgepedia.org/thue-siegel-roth-theorem), Shidlovsky's work on E-functions, and Sprindžuk's solution of the Mahler conjecture[11](https://www.cambridge.org/core/books/transcendental-number-theory/21DDE8BFF5B527AF751D744736CE1CF2).
## Known and unknown examples
The proved list includes e, π, log 2, e√2, eπ, 2√2, and Γ(1/4)[4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs). The open list is startling to non-specialists: e + π, eπ, log π, πe, Γ(1/5), ζ(3), and the Euler constant γ are each not known to be transcendental[4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs). For π + e and π·e at least one of the two is transcendental (probably both), since both would be algebraic only if a certain quadratic had algebraic roots, but transcendence has not been proven for either number on its own[9](https://mathworld.wolfram.com/TranscendentalNumber.html).
**Zeta values.** Apéry proved in 1978 that ζ(3) = 1.202 056 903 159 594 285 399 738 161 511… is irrational, but its transcendence is not known; ζ(5) = 1.036 927 755 143 369 926 331 365 486 457…[4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs). After Apéry's proof, his approach has been widely developed for irrationality proofs of zeta and multizeta values[10](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Strategy.pdf).
## By the numbers: measure, cardinality, and approximation
**Counting versus constructing.** Cantor's 1874 paper shows the real algebraic numbers can be enumerated in a sequence indexed by the natural numbers[12](http://uamte.math.byu.edu/~grant/courses/cantor1874.pdf). Since no sequence of real numbers can enumerate all numbers in an open interval, every nonempty open interval contains at least one transcendental number; where Liouville constructed a transcendental number, indeed a whole family of them, Cantor used a counting argument to show the reals must include many[13](https://www.cs.uwyo.edu/~ruben/static/pdf/cantor-trio-final.pdf). Cantor thereby proved that the transcendental real numbers form a set of the cardinality of the continuum[1](https://encyclopediaofmath.org/wiki/Transcendental_number).
**Measure.** Borel, introducing the first concepts of measure theory, established that almost all real numbers are transcendental[1](https://encyclopediaofmath.org/wiki/Transcendental_number). The Liouville numbers illustrate how thin an explicit class can be: they form an everywhere-dense subset of the real axis of continuum cardinality but zero [Lebesgue measure](https://www.edgepedia.org/lebesgue-measure)[1](https://encyclopediaofmath.org/wiki/Transcendental_number), and every real number can be represented as a sum, and also as a product, of two Liouville numbers[3](https://simonrs.com/eulercircle/irpw2022/varun-transcendental-paper.pdf).
**Approximation and transcendence.** Liouville's inequality says that if f is a minimal polynomial of degree d ≥ 2 for ρ, then there exists c(ρ) > 0 such that \|ρ − p/q\| > c(ρ)/qd for all rationals p/q[5](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_Number_Theory_(Veerman)/01%3A_A_Quick_Tour_of_Number_Theory/1.03%3A_Algebraic_and_Transcendental_Numbers). A real number ρ is a Liouville number if for every n there is a rational p/q with \|ρ − p/q\| < 1/qn; such numbers violate the inequality for every d and are therefore all transcendental[5](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_Number_Theory_(Veerman)/01%3A_A_Quick_Tour_of_Number_Theory/1.03%3A_Algebraic_and_Transcendental_Numbers). Roth's theorem (1955) closes the gap from above: if α is algebraic then its irrationality measure µ(α) = 2, for which Roth received the Fields Medal[5](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_Number_Theory_(Veerman)/01%3A_A_Quick_Tour_of_Number_Theory/1.03%3A_Algebraic_and_Transcendental_Numbers).
**Irrationality measures in practice.** Exact values are known for Liouville's constant (µ = ∞) and µ(e) = 2[14](https://mathworld.wolfram.com/IrrationalityMeasure.html). For π, the first upper bound, 42, was proved by [Kurt Mahler](https://www.edgepedia.org/kurt-mahler) in 1953, improved by Mignotte (1974), Chudnovsky (1982), and Hata (1990, 1993), then by Salikhov to 7.606308, and by Zeilberger and Zudilin (2020) to 7.103205334137…[15](https://msp.org/cnt/2020/9-4/moscow-v9-n4-p06-s.pdf). Other upper bounds include µ(ζ(5)) ≤ 5.09541179 (Zudilin 2013) and µ(ln 2) ≤ 3.57455391 (Marcovecchio 2009)[14](https://mathworld.wolfram.com/IrrationalityMeasure.html).
## How it compares: irrational, algebraic, transcendental
Every transcendental number is irrational, but not conversely[2](https://www.worldscientific.com/worldscibooks/10.1142/q0085). Roth's theorem is essentially the best possible result of its kind, since Dirichlet's approximation theorem shows the ε = 0 case fails[3](https://simonrs.com/eulercircle/irpw2022/varun-transcendental-paper.pdf). Roth's theorem does allow proving transcendence of a few more numbers, such as the [Champernowne constant](https://www.edgepedia.org/champernowne-constant) 0.123456789101112…, but even now the vast majority of transcendental numbers remain unproven as such[3](https://simonrs.com/eulercircle/irpw2022/varun-transcendental-paper.pdf).
The gap between the measure-theoretic statement and explicit knowledge is sharpest at the digit level. For no concrete triple (b, a, x)—with b ≥ 3 an integer, a a digit in {0,…,b−1}, and x an algebraic irrational—has anyone so far demonstrated that the digit a appears infinitely often in the base-b expansion of x; a stronger conjecture, also due to Borel, holds that algebraic irrational reals are normal, with every block of n digits in base b appearing with frequency 1/bn • [4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs).
## Generalizations and conjectures
**Schanuel's conjecture** is the field's central conjecture: if x1,…,xn are Q-linearly independent complex numbers, then at least n of the 2n numbers x1,…,xn, ex1,…,exn are algebraically independent[4](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs). So far, the only proposed strategy that would yield Schanuel's conjecture is the one outlined by Daniel Roy in 2001; the main open problem of algebraic independence of logarithms of algebraic numbers reduces, according to Roy, to estimating the rank of matrices whose entries are logarithms of algebraic numbers[10](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Strategy.pdf). Alongside Schanuel, the constant problem ranks among the fundamental outstanding problems of transcendental number theory[9](https://mathworld.wolfram.com/TranscendentalNumber.html).
## Recent developments
**Irrationality measures.** The irrationality exponent of ζ(3) was improved to µ(ζ(3)) < 5.5138800, exactly 5.51387997400447264…, beating the Rhin–Viola bound of 5.513891 from 2001, which had stood for twenty-five years; the chain of prior records ran from Apéry's own 13.41782… through Hata's 7.377956…[16](https://arxiv.org/html/2609.26980). For π, Bai obtained µ(π) ≤ 7.101862832357 by displacing the parameter point of Salikhov's construction inside a narrow cone, a further step past Zeilberger–Zudilin's 7.103205334137[16](https://arxiv.org/html/2609.26980) • [15](https://msp.org/cnt/2020/9-4/moscow-v9-n4-p06-s.pdf).
**Formalization and new independence results.** A complete formalization of the Gelfond–Schneider theorem in Lean 4 was produced in 2026, to the authors' knowledge the first in any theorem prover, including a formal derivation of the transcendence of the Gelfond–Schneider constant √2√2 • [7](https://arxiv.org/pdf/2603.24823). The Hermite–Gelfond–Schneider method has also been used very recently: in 2026 for new algebraic-independence results involving values of Weierstrass elliptic and zeta functions, and in 2025 toward generalizing the Stéphanois Theorem to Igusa invariants of genus-two curves[10](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Strategy.pdf).
## References
1. [Transcendental number, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Transcendental_number)
2. [Periods and Special Functions in Transcendence, World Scientific](https://www.worldscientific.com/worldscibooks/10.1142/q0085)
3. [A Historical Introduction to Transcendental Number Theory](https://simonrs.com/eulercircle/irpw2022/varun-transcendental-paper.pdf)
4. [Transcendental Number Theory: recent results and open problems, M. Waldschmidt](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs)
5. [Algebraic and Transcendental Numbers, LibreTexts](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_Number_Theory_(Veerman)/01%3A_A_Quick_Tour_of_Number_Theory/1.03%3A_Algebraic_and_Transcendental_Numbers)
6. [Algebraic Versus Transcendental Objects, Britannica](https://www.britannica.com/topic/algebraic-versus-transcendental-objects-1084407)
7. [A formalization of the Gelfond–Schneider theorem in Lean 4, arXiv](https://arxiv.org/pdf/2603.24823)
8. [Lindemann theorem, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Lindemann_theorem)
9. [Transcendental Number, Wolfram MathWorld](https://mathworld.wolfram.com/TranscendentalNumber.html)
10. [An introduction to the strategy of transcendence proofs, M. Waldschmidt](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Strategy.pdf)
11. [Transcendental Number Theory, A. Baker, Cambridge University Press](https://www.cambridge.org/core/books/transcendental-number-theory/21DDE8BFF5B527AF751D744736CE1CF2)
12. [On a Property of the Class of all Real Algebraic Numbers, G. Cantor (1874, translation)](http://uamte.math.byu.edu/~grant/courses/cantor1874.pdf)
13. [Cantor's 1874 proof of the existence of transcendental numbers](https://www.cs.uwyo.edu/~ruben/static/pdf/cantor-trio-final.pdf)
14. [Irrationality Measure, Wolfram MathWorld](https://mathworld.wolfram.com/IrrationalityMeasure.html)
15. [The irrationality measure of π is at most 7.103205334137…, Zeilberger & Zudilin, Confluentes Mathematici](https://msp.org/cnt/2020/9-4/moscow-v9-n4-p06-s.pdf)
16. [A new upper bound for the irrationality exponent of ζ(3), arXiv](https://arxiv.org/html/2609.26980)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Algebraic number theory › Number fields and algebraic integers*
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