Is the following number rational or irrational?

Question

Consider the following number:

1/(2)^(1/2) - i*(arcsin(1.235711131719...)) - pi/2) = (a),
which is approximately equal to .03332266499855....
where i is the square root of -1, the decimal digits in the argument of the arcsine function consists of the string of all the prime numbers, and pi is defined to be the ratio of the circumference of a circle to its diameter. Is this number rational or irrational? Or, if one takes the first 100 primes and strings them together approximate the number in the argument of the arcsine function, can someone calculate the number of (a) to 100 digits accuracy?

Inquisitively,

Edwin G. Schasteen

Answer 1

dude, you're taking the arcsine of a number that's bigger than 1. its undefined.

edit:
actually, it's a little more complicated than that, since if you're working over complex numbers, arcsine is defined over the whole plane (although it is multivalued). I found the following formula:
i*arcsin(z) = log(iz + sqrt(1-z^2))

here, z is a real number larger than 1, so that square root is going to be imaginary. we can rewrite it as
log(i(z + sqrt(z^2-1)),
where it's now clear that we're taking the log of a purely imaginary number.

using the formula log(w) = ln|w| + i*arg(w), we get that
log(i(z + sqrt(z^2-1)) = ln(z + sqrt(z^2-1)) + i*pi/2.

oh wow, so that cancels with the i*pi/2 in your formula, and you're left with the formula
(a) = 1/(2)^(1/2) - ln(z + sqrt(z^2-1)),
which is manifestly real and well-defined.

this number built up out of all the primes is pretty crazy, so sorry I can't help you decide if its rational, but the formula I just wrote should be useful for getting the approximation you wanted.

Answer 2

Momolala is right on. Sorry you wasted your time.