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). In 1900, David Hilbert posed a question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is a b necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. The transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. But since i is algebraic, π therefore must be transcendental. Then, since e i π = −1 is algebraic (see Euler's identity), i π must be transcendental. He first proved that e a is transcendental if a is a non-zero algebraic number. In 1882, Ferdinand von Lindemann published the first complete proof of the transcendence of π. Ĭantor's work established the ubiquity of 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. He also gave a new method for constructing transcendental numbers. In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was e, by Charles Hermite in 1873. Liouville showed that all Liouville numbers are transcendental. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers are called Liouville numbers, named in his honour. In other words, the nth digit of this number is 1 only if n is one of the numbers 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. In which the nth digit after the decimal point is 1 if n is equal to k! ( k factorial) for some k and 0 otherwise. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x 2 − 2 = 0. Hence, the set of real numbers consists of non-overlapping rational, algebraic non-rational and transcendental real numbers. The converse is not true: Not all irrational numbers are transcendental. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. Though only a few classes of transcendental numbers are known – partly because it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. The best known transcendental numbers are π and e. In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients.
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