Christoffer De Geer March 20, 2026

What is Liouville's Constant?

Ones in Liouville's constant: exponentially sparse

A 1 appears at positions 1, 2, 6, 24, 120, 720... (the factorials). All other positions are 0. The gaps grow exponentially: after position 24 the next 1 is at position 120.

Methods for proving transcendence: Liouville opened the door

Each breakthrough opened a new tool for proving numbers transcendental. Lindemann proved π is transcendental in 1882, ending the squaring-the-circle problem.

Related topics

Transcendental Numbers Champernowne Irrational Numbers

Champernowne Constant → Transcendental Numbers → Irrational Numbers → Continued Fractions →

Key facts about Liouville's Constant

Liouville's constant L = 0.110001000000000000000001... has 1s at positions 1!, 2!, 3!, 4!, ... and 0s elsewhere. Joseph Liouville constructed it in 1844 as the first explicit transcendental number, 29 years before Hermite proved e transcendental. His proof showed algebraic numbers cannot be approximated by rationals too accurately: the rapidly spreading 1s in L violate this bound. The construction elegantly demonstrated transcendentals exist without Cantor's later diagonal argument.

Used in

∑Mathematics

✓

⚛Physics

–

⚙Engineering

–

🧬Biology

–

💻Computer Sci

–

📊Statistics

–

📈Finance

–

🎨Art

–

🏛Architecture

–

♪Music

–

🔐Cryptography

–

🌌Astronomy

–

⚗Chemistry

–

🦉Philosophy

✓

🗺Geography

–

🌿Ecology

–

Want to test your knowledge?

Question

What proportion of real numbers are transcendental?

tap · space

1 / 10

Ready to play?

Pi

Memorize pi, e, and 38 mathematical constants using the numpad path method

Play now

Works on any device.

Topic roundups

#memoryAll memory gamesHold a sequence, grid, or list in working memory and reproduce it under pressure.#numbersAll numbers gamesArithmetic, ratios, units, and number sense in short timed rounds.#sequenceAll sequence gamesReproduce or extend an ordered string of items in the right order.