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  2. Category:Irrational numbers - Wikipedia

    en.wikipedia.org/wiki/Category:Irrational_numbers

    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.

  3. Irrational number - Wikipedia

    en.wikipedia.org/wiki/Irrational_number

    Examples are e r and π r, which are transcendental for all nonzero rational r. Because the algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3 π + 2, π + √ 2 and e √ 3 are irrational (and even transcendental).

  4. Irrationality measure - Wikipedia

    en.wikipedia.org/wiki/Irrationality_measure

    Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...

  5. Proof that e is irrational - Wikipedia

    en.wikipedia.org/wiki/Proof_that_e_is_irrational

    This last fact implies that e 4 is irrational. His proofs are similar to Fourier's proof of the irrationality of e. In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that e is not a root of a third-degree polynomial with rational coefficients, which implies that e 3 is irrational. [12]

  6. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that is a rational number. This proof uses that 2 {\displaystyle {\sqrt {2}}} is irrational (an easy proof is known since Euclid ), but not that 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is irrational (this is true, but the proof ...

  7. Completeness of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Completeness_of_the_real...

    The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom.Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are non Archimedean fields ...

  8. Proof of impossibility - Wikipedia

    en.wikipedia.org/wiki/Proof_of_impossibility

    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.

  9. Proof by contradiction - Wikipedia

    en.wikipedia.org/wiki/Proof_by_contradiction

    The classic proof that the square root of 2 is irrational is a refutation by contradiction. [11] Indeed, we set out to prove the negation ¬ ∃ a, b ∈ . a/b = √ 2 by assuming that there exist natural numbers a and b whose ratio is the square root of two, and derive a contradiction.