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More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, [2] and reductio ad impossibile.
Written in 1873, this proof uses the characterization of as the smallest positive number whose half is a zero of the cosine function and it actually proves that is irrational. [3] [4] As in many proofs of irrationality, it is a proof by contradiction.
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.
Hippasus, engraving by Girolamo Olgiati, 1580. Hippasus of Metapontum (/ ˈhɪpəsəs /; Greek: Ἵππασος ὁ Μεταποντῖνος, Híppasos; c. 530 – c. 450 BC) [1] was a Greek philosopher and early follower of Pythagoras. [2][3] Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the ...
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 ...
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.
In proof by contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that is an irrational number:
One proof of the number's irrationality is the following proof by infinite descent. It is also a proof of a negation by refutation : it proves the statement " 2 {\displaystyle {\sqrt {2}}} is not rational" by assuming that it is rational and then deriving a falsehood.