The Real Magic Behind Imaginary Numbers

It turns out, by solving one impossible equation, we solve all the algebraic equations.

We’ve all learned in basic algebra that the square of any number is positive. So the equation

Has no solution. Should we just give up? Not so fast, mathematics is a creative discipline, so we can just invent a solution, i, so that

This imaginary unit, turns out, has super powers. By allowing it to solve this simple equation, it actually enables us to solve all algebraic equations involving any polynomials (and in fact to an even broader class of functions). This power is unleashed by the deep connections between geometry and algebra, enabled through complex numbers.

Below are my physicist intuitions on this connection.

Factoring Distance

The first link comes from rethinking the formula for distance (or the square of distance). Imagine I walk 3 meters forward and 4 meters rightward, how far have I walked? The answer is 5 meters and comes from the Pythagorean theorem, since 5² = 3² + 4². More generally, we have the well known formula:

What does the imaginary unit, i, have to do with this? The answer is factorization. There is a well known formula (a + b)(a − b) = a² − b², but it’s just missing a sign when comparing to…