Why imaginary number, i, seems so imaginary?

The imaginary number, i, is introduced into mathematics rather late. Even then, it was called imaginary, not real. We now know imaginary number is very real in the real world. Why does imaginary number, i, seem so imaginary to us?

We can express a complex number in two parts, its magnitude and its phase. Magnitude is represented by a real number and phase is represented by an imaginary number. We all know what a magnitude is. But we have a much less intuition about phase. This is probably because our senses have much less capacity to process phase. Our hearing can detect phase of low frequency sound. That is how we can detect the direction of sound by determining the phase differences of sounds received by two different ears. But we can’t distinguish the phases of high frequency sounds, probably because the phase difference of high frequency sounds occurs at a much shorter time period. Light waves have much higher frequencies than sound waves. We can’t detect phases of light waves.

We have very limited sense of phases. Probably this contributed to our less developed intuition about the imaginary number, which, among other things, is a representation of phase. Similarly, most equations in physics contain only real values. An equation containing imaginary number, such as Schrodinger’s equation, looks unnatural to us.

Fourier transform is a very fundamental tool in our understanding of nature. It transforms time information into frequency information. Only when phase information is preserved, or when imaginary numbers are retained in Fourier transform, Fourier transform is reversible, or total information is preserved.

With imaginary number, one more dimension is added to the number system. As a result, analytical theories can be presented in much more compact forms. This is especially helpful for modelling complex issues, such as in quantum mechanics.