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  2. e (mathematical constant) - Wikipedia

    en.wikipedia.org/wiki/E_(mathematical_constant)

    The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .

  3. Euler numbers - Wikipedia

    en.wikipedia.org/wiki/Euler_numbers

    The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. The latter is the function in the definition. They also occur in combinatorics , specifically when counting the number of alternating permutations of a set with an even number of elements.

  4. Proof that e is irrational - Wikipedia

    en.wikipedia.org/wiki/Proof_that_e_is_irrational

    The number e was introduced by Jacob Bernoulli in 1683. More than half a century later, Euler , who had been a student of Jacob's younger brother Johann , proved that e is irrational ; that is, that it cannot be expressed as the quotient of two integers.

  5. Euler's constant - Wikipedia

    en.wikipedia.org/wiki/Euler's_constant

    Using the same approach, in 2013, M. Ram Murty and A. Zaytseva showed that the generalized Euler constants have the same property, [3] [44] [45] where the generalized Euler constant are defined as = (= ⁡ = ()), where ⁠ ⁠ is a fixed list of prime numbers, () = if at least one of the primes in ⁠ ⁠ is a prime factor of ⁠ ⁠, and ...

  6. Project Euler - Wikipedia

    en.wikipedia.org/wiki/Project_Euler

    The first Project Euler problem is Multiples of 3 and 5. If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000. It is a 5% rated problem, indicating it is one of the easiest on the site.

  7. Contributions of Leonhard Euler to mathematics - Wikipedia

    en.wikipedia.org/wiki/Contributions_of_Leonhard...

    Euler invented the calculus of variations including its most well-known result, the Euler–Lagrange equation. Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory.

  8. Euler's identity - Wikipedia

    en.wikipedia.org/wiki/Euler's_identity

    Euler's identity is a special case of Euler's formula, which states that for any real number x, e i x = cos ⁡ x + i sin ⁡ x {\displaystyle e^{ix}=\cos x+i\sin x} where the inputs of the trigonometric functions sine and cosine are given in radians .

  9. List of topics named after Leonhard Euler - Wikipedia

    en.wikipedia.org/wiki/List_of_topics_named_after...

    Euler numbers, integers occurring in the coefficients of the Taylor series of 1/cosh t; Eulerian numbers count certain types of permutations. Euler number (physics), the cavitation number in fluid dynamics. Euler number (algebraic topology) – now, Euler characteristic, classically the number of vertices minus edges plus faces of a polyhedron.

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