Transcendental numbers

Did we look at these?

  1. pi = 3.1415 …
  2. e = 2.718 …
  3. Euler’s constant, gamma = 0.577215 … = lim n -> infinity > (1 + 1/2 + 1/3 + 1/4 + … + 1/n – ln(n)) (Not proven to be transcendental, but generally believed to be by mathematicians.)
  4. Catalan’s constant, G = sum (-1)^k / (2k + 1 )^2 = 1 – 1/9 + 1/25 – 1/49 + … (Not proven to be transcendental, but generally believed to be by mathematicians.)
  5. Liouville’s number 0.110001000000000000000001000 … which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere.
  6. Chaitin’s “constant”, the probability that a random algorithm halts. (Noam Elkies of Harvard notes that not only is this number transcendental but it is also incomputable.)
  7. Chapernowne’s number, 0.12345678910111213141516171819202122232425… This is constructed by concatenating the digits of the positive integers. (Can you see the pattern?)
  8. Special values of the zeta function, such as zeta (3). (Transcendental functions can usually be expected to give transcendental results at rational points.)
  9. ln(2).
  10. Hilbert’s number, 2(sqrt 2 ). (This is called Hilbert’s number because the proof of whether or not it is transcendental was one of Hilbert’s famous problems. In fact, according to the Gelfond-Schneider theorem, any number of the form ab┬áis transcendental where a and b are algebraic (a ne 0, a ne 1 ) and b is not a rational number. Many trigonometric or hyperbolic functions of non-zero algebraic numbers are transcendental.)
  11. epi

2 comments for “Transcendental numbers

  1. dearieme
    March 21, 2019 at 16:08

    And yet e to the power (i x pi) is not transcendental. It’s not even irrational. Neither is it imaginary, nor complex.

    It’s a miracle I tell ‘ee.

    You can use a hell of a lot of maths without meeting anything on that list bar 1, 2, 3, and 9.

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