Did we look at these?

- pi = 3.1415 …
- e = 2.718 …
- 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.)
- 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.)
- Liouville’s number 0.110001000000000000000001000 … which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere.
- 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.)
- Chapernowne’s number, 0.12345678910111213141516171819202122232425… This is constructed by concatenating the digits of the positive integers. (Can you see the pattern?)
- Special values of the zeta function, such as zeta (3). (Transcendental functions can usually be expected to give transcendental results at rational points.)
- ln(2).
- 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 a^{b} is 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.)
- e
^{pi}

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.

I defer to your expertise here.