Does an equation with i as an exponent have i solutions?

Answer 1

Hi,

Simply put, any single mathematical equation either has an integer (whole number) number of solutions or infinitely many solutions, regardless of the specifics of the equation. There is not really a sense in which an equation can have "i" solutions, in just the same way that there is not really a sense in which an equation can have 3.5, or π solutions.

With that said, I believe what your question is regarding is the number of solutions for x in an equation of the form xⁱ = a, for some given number "a". The theorem I believe you are hoping to reference here is called the Fundamental Theorem of Algebra, which states that any polynomial of degree n (an integer) will have n complex zeros. The issue in trying to apply this theorem here is that "i" is not an integer, so this theorem will not help us to determine the number of solutions in any way (the same way it wouldn't be of help if I wanted to know the number of complex solutions to x^π = a). (For technical reasons, the theorem will actually help as long as the "p" in x^p = a is a rational number, but that's not really of concern here).

If I am correct that your question is really about finding the number of complex solutions to an equation like xⁱ = a, there are still methods to find this out. The first step will be to recognize that xⁱ = e^i·ln(x). We can then apply Euler's formula, which states that e^iθ = cos(θ) + i·sin(θ), so we see that if xⁱ = a, this is equivalent to cos(ln(x)) + i·sin(ln(x)) = a. If the number "a" we started with is a real number, then we are guaranteed it has complex part 0, and so sin(ln(x)) = 0, indicating ln(x) must be an integer multiple of π, from which we can conclude that cos(ln(x)) is either 1 or -1. So the only real numbers "a" for which the equation xⁱ = a has any solutions are -1 and 1, and in this case it will have infinitely many solutions. If "a" is allowed to be complex, more work is required. We can make a similar argument if "a" is purely imaginary (real part is 0), but otherwise it is not immediately clear whether or not solutions exist, and how many there are.

Thanks,

Aaron