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Euler's proof. Euler wrote the first proof of the fact that e is irrational in 1737 (but the text was only published seven years later). [1][2][3] He computed the representation of e as a simple continued fraction, which is. Since this continued fraction is infinite and every rational number has a terminating continued fraction, e is irrational.
v. t. e. 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.
Irrational number. The number √ 2 is irrational. In mathematics, the irrational numbers (in- + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also ...
Apéry's theorem. In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ (3) is irrational. That is, the number. cannot be written as a fraction where p and q are integers. The theorem is named after Roger Apéry. The special values of the Riemann zeta function at even integers ( ) can be shown in ...
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers and , with , there exist integers and such that and. Here represents the integer part of . This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of ...
In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0° ≤ θ ≤ 90° for which the sine of θ degrees is also a rational number are: [1] In radians, one would require that 0 ≤ x ≤ π /2, that x / π be rational, and that sin x be rational. The conclusion is then that the ...
Liouville number. In number theory, a Liouville number is a real number with the property that, for every positive integer , there exists a pair of integers with such that. The inequality implies that Liouville numbers possess an excellent sequence of rational number approximations. In 1844, Joseph Liouville proved a bound showing that there is ...
In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m, n such that. The condition that ξ is irrational cannot be omitted. Moreover the constant is the best possible; if we replace ...
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