One may ask after reading the Euler's formula article, why is the imaginary unit, $i$, so important in physics? The answer lies in the wave-particle duality of light and matter, which is fundamentally described by complex numbers and their properties.

With the wave-particle duality of light and matter, the double-slit experiment of Young (1801) demonstrated the fundamental principles of quantum mechanics, though it was scoped to optical phenomena. The interference pattern of charged matter passing through two slits (such as the planes of a crystal) is a result of the wave-like behavior of charged particles, which can be described by a mathematical function known as a wave function.

A theorem by Mertzbacher (1961) shows that if the wave is phase invariant, that is, if the wave function, $f(x)$, is effectively unchanged by a phase shift of $\epsilon$ to the argument, then the square-root of minus one is a required property, providing a direct route from the phase invariance of physical systems to their inherent complexity.