A 1 appears at decimal positions 1, 2, 6, 24, 120, 720, 5040… (the factorials), and 0 everywhere else. The gaps grow exponentially: the next 1 after position 24 is at position 120.
The truncated series 0.11, 0.110001, 0.1100010000000000000000000001… are rational approximations to L that are far more accurate than any algebraic number allows. This violates Liouville's theorem for algebraic numbers, so L cannot be algebraic.
Liouville's constant L = 0.1100010000000000000000000100… is defined by placing the digit 1 at positions 1!, 2!, 3!, 4!, 5!,… in the decimal expansion and 0 everywhere else. The 1s appear at positions 1, 2, 6, 24, 120, 720,… becoming exponentially more spread out.
In 1844, Joseph Liouville proved this number is transcendental, making it the first specific number ever proved to be transcendental. His proof used the Liouville approximation theorem: algebraic numbers of degree n cannot be too well approximated by rationals. Liouville showed that L can be approximated extraordinarily well by rationals, which proved it cannot be algebraic.
The proof came 39 years before Hermite proved e is transcendental (1873) and 38 years before Lindemann proved π is transcendental (1882). Liouville's construction was deliberately artificial to make the proof work, but it opened the door to all of transcendence theory. His theorem implies that almost all real numbers are transcendental.
Liouville's 1844 proof showed transcendental numbers exist but used a method too limited for natural constants like e and π. Hermite (1873) and Lindemann (1882) needed completely different approaches. Liouville's work was the first step in a theory that now has hundreds of methods.
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.