The Long Run Guess

[random 8]

Is it related to Pi?

Nope.

This may have been before i understood the polar form of complex numbers. π does show up in the number if written in the form re^iθ. Apologies for any confusion.

abs(a),abs(b) ≤ 100

Yes.

φ∈{±atan(x^±1):x∈{5/12, 7/24, 8/15, 9/40, 11/60, 12/35, 13/84, 16/63, 20/21, 20/99, 28/45, 33/56, 36/77, 39/80, 48/55, 60/91, 65/72}}

φ is none of those numbers.

(since you wrote in a previous answer, b isn't the irrational part, a is irrational)

I said the number is not of the form a*i, where a is rational or real. That means, in a+bi, where a,b∈ℝ, one or both of a and b is irrational.