> but didn't we just invent complex numbers because they are useful?
Not any more than we invented natural numbers because they are useful.
There are several ways to naturally derive complex numbers, either from mathematics or from physics.
For one, complex numbers are probably the simplest possible extension of the real numbers in which all real-valued polynomials have roots (for example, x^2 + 1 doesn't have a root if x has to be real). This is the same reason why the non-transcendental irrational numbers were invented (such as sqrt(2) ).
(Incidentally, the transcendental numbers (pi, e) are less justified than the complex numbers from this point of view - any polynomial of any rank whose coefficients are non-transcendental real numbers has roots that are either real non-transcendental numbers, or a complex number whose real and imaginary parts are real non-transcendental numbers )
For a physical explanation, complex numbers are the best way we know of describing wave mechanics (either classical or quantum), and in general periodic phenomena and how they compose.
Not any more than we invented natural numbers because they are useful.
There are several ways to naturally derive complex numbers, either from mathematics or from physics.
For one, complex numbers are probably the simplest possible extension of the real numbers in which all real-valued polynomials have roots (for example, x^2 + 1 doesn't have a root if x has to be real). This is the same reason why the non-transcendental irrational numbers were invented (such as sqrt(2) ).
(Incidentally, the transcendental numbers (pi, e) are less justified than the complex numbers from this point of view - any polynomial of any rank whose coefficients are non-transcendental real numbers has roots that are either real non-transcendental numbers, or a complex number whose real and imaginary parts are real non-transcendental numbers )
For a physical explanation, complex numbers are the best way we know of describing wave mechanics (either classical or quantum), and in general periodic phenomena and how they compose.