h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Imaginary numbers (coefficient i or j = sqrt(-1)) are only half the story of complex numbers of the form a + bi, where a is the real part and bi is the imaginary part, comprising b, the coefficient, multiplied by sqrt(-1).

The motivation for complex numbers in engineering and physics had to do with the concept of potential energy. You can't actually "see" the potential energy that results from compression or expansion of a mechanical spring, for example, so we needed a way to express such quantities in a vector form that assures that the real part (kinetic energy) doesn't get mixed with terms involving the potential energy.

"Imaginary" numbers are no more or less a product of imagination than are the real numbers.

If you were to ask an artificial intelligence based on current computational engines what a number was (a sort of Turing test), does anyone think it could render an intelligible answer? Probably not.

If you were to ask someone with an ordinary human brain what exactly a "length", or equivalently, a "time interval" actually is, do you think anyone could give an intelligible answer? Again, no. Newton could not. Einstein could not. They could only tell you how things move with respect to them, not what the concepts actually meant. This is the best evidence that even a human mind has fundamental limitations in terms of what is loosely referred to as intelligence. And most human minds never even stretch even a fraction as far as those did.

Feeling any better about imaginary numbers now?