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In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, [2] and reductio ad impossibile. [3] A mathematical proof employing proof by contradiction usually proceeds as follows: The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies ...
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
In mathematics, the irrational numbers ... Here is a proof by contradiction that log 2 3 ... there is a second definition of an irrational number used in ...
A mathematical proof is a deductive argument for ... is an irrational number. To ... this takes the form of a proof by contradiction in which the nonexistence of the ...
One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.
However, indirect methods such as proof by contradiction can also be used with contraposition, as, for example, in the proof of the irrationality of the square root of 2. By the definition of a rational number , the statement can be made that " If 2 {\displaystyle {\sqrt {2}}} is rational, then it can be expressed as an irreducible fraction ".
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.
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.