3 posts in mathematics
How I wish complex numbers were taught
In his 1946 essay «The metaphysical principles of the infinitesimal calculus», the philosopher René Guénon asserts that any number wich is not a positive integer is not an actual number. His argumentation rises from noticing that natural numbers, the first to be introduced (or discovered, but there may be a whole other chapter of discussion around which is the correct verb) by humans, contain the purest idea of «quantity». Two serves to indicate a precise quantity of objects, and there is no other characteristic that we can associate with the number two.

Another intuitive approach: Euler's formula for complex numbers

Complex numbers do exist.

That said, I keep meeting people who think imaginary numbers are just an artificial instrument to solve differential equations: you get a $\sqrt{-1}$ somewhere, pretend it is something instead of a nonsensical symbol, do your calculations and take the real part of the result. And even though the true mathematicians now might be struggling with hyperreal or transfinite numbers, we normal people are still having trouble with the good old $i$.

Much of the sense of mistery imaginary numbers carry with themselves is related with the esoteric aura surrounding Euler’s identity:

Understanding how the exponential function works is like putting the last piece in the middle of a puzzle: everything in calculus and algebra makes sense. Not only do a lot of things in real and complex analysis come from the very definition of Euler’s number and its consequent magic properties, but also things everybody uses in his mathematical life, like taking a real number-power of something, have their ultimate definition in $e^x$.
For example, how do you calculate $2^\sqrt{2}$ or $3^\pi$ without knowing how the power operation with a non-rational exponent is defined?