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A more recent proof by Wadim Zudilin is more reminiscent of Apéry's original proof, [6] and also has similarities to a fourth proof by Yuri Nesterenko. [7] These later proofs again derive a contradiction from the assumption that ζ ( 3 ) {\displaystyle \zeta (3)} is rational by constructing sequences that tend to zero but are bounded below by ...
In the Middle Ages, the development of algebra by Muslim mathematicians allowed irrational numbers to be treated as algebraic objects. [19] Middle Eastern mathematicians also merged the concepts of " number " and " magnitude " into a more general idea of real numbers , criticized Euclid's idea of ratios , developed the theory of composite ...
In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Lambert (1761) gave a flawed proof that π cannot be rational; Legendre (1794) completed the proof [11] and showed that π is not the square root of a rational number. [12]
In the 1760s, Johann Heinrich Lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction /, where and are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus .
Later, in 1891, Cantor used his more familiar diagonal argument to prove the same result. [17] While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number, [18] [19] the proofs in both the aforementioned papers give methods to construct transcendental numbers. [20]
The irrationality exponent or Liouville–Roth irrationality measure is given by setting (,) =, [1] a definition adapting the one of Liouville numbers — the irrationality exponent () is defined for real numbers to be the supremum of the set of such that < | | < is satisfied by an infinite number of coprime integer pairs (,) with >.
The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem.Charles Hermite first proved the simpler theorem where the α i exponents are required to be rational integers and linear independence is only assured over the rational integers, [4] [5] a result sometimes referred to as Hermite's theorem. [6]
The following 1953 proof by Dov Jarden has been widely used as an example of a non-constructive proof since at least 1970: [4] [5] CURIOSA 339. A Simple Proof That a Power of an Irrational Number to an Irrational Exponent May Be Rational. is either rational or irrational. If it is rational, our statement is proved.