numbers, named in his honour. Liouville showed that all Liouville numbers are transcendental. The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was , by Charles Hermite in 1873. In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers. Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers. Cantor's work established the ubiquity of transcendental numbers. In 1882, Ferdinand von Lindemann published the first complete proof of the transcendence of . He first proved that is transcendental if is a non-zero algebraic number. Then, since is algebraic (see Euler's identity), must be transcendental. But since is algebraic, therefore must be transcendental. This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. The transcendence of allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If is an algebraic number that is not zero or one, and is an irrational algebraic number, is necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers). Properties A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be irrational, since a rational number is the root of an integer polynomial of degree one. The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both subsets to be countable. This makes the transcendental numbers uncountable. No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers