wiki Imaginary Numbers Are Reality - Nautilus
Euler identity:
To some, this is a near-mystical formula. Here we have the base of natural logarithms e; the numbers 0 and 1, which are both unique cases on the whole number line; the imaginary number, a special case all of its own; and π, which as we know is a source of power in mathematics. Despite being discovered at different times by different people looking at different pieces of mathematics, it turns out they are interrelated, coexisting in this elegant, simple equation. Seen from a slightly different perspective, perhaps we shouldn’t be surprised. As with π itself, there really isn’t anything mystical about this formula. It results from the fact that numbers change and transform themselves and each other through rotations. That only happens because of what numbers are: representations of the relationships between quantities. We don’t find anything mystical about moving along the familiar “real” number line by adding and subtracting. And there’s nothing different, really, about the transformations that come about through multiplications and divisions. Remember that sines and cosines are just ratios—one number divided by another—that are related to the angles within triangles, and you can represent those angles as fractions or multiples of π in units known as radians. So what we’re discovering here is not some deep mystery about the universe, but a clear and useful set of relationships that are a consequence of defining numbers in various different ways.