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Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a b is rational: [28] [29] Consider √ 2 √ 2; if this is rational, then take a = b = √ 2. Otherwise, take a to be the irrational number √ 2 √ 2 and b = √ 2. Then a b = (√ 2 √ 2) √ 2 = √ 2 √ 2 · √ 2 = √ 2 2 = 2 ...
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 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]
A more general proof shows that the mth root of an integer N is irrational, unless N is the mth power of an integer n. [7] That is, it is impossible to express the m th root of an integer N as the ratio a ⁄ b of two integers a and b , that share no common prime factor , except in cases in which b = 1.
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. The golden ratio (denoted or ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x 2 − x − 1 = 0.
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.
In mathematics, an irrational number is any real number that is not a rational number, i.e., one that cannot be written as a fraction a / b with a and b integers and b not zero. This is also known as being incommensurable, or without common measure. The irrational numbers are precisely those numbers whose expansion in any given base (decimal ...
This shows that any irrational number has irrationality measure at least 2. The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2.