No real number, squared, can ever be negative. And yet mathematicians started from the imaginative question "what if a number like that did exist?" and built an entirely new number system out of it. Step into the world of imaginary and complex numbers — used today in everything from electrical engineering to quantum mechanics (some of this content is high-school level).
Define i as the number whose square is −1 (i² = −1). A real number multiplied by i is called an imaginary number, and a number of the form a+bi, a real number plus an imaginary number, is called a complex number. A complex number can be drawn as a single point on a plane (the complex plane) where the horizontal axis is the real part and the vertical axis is the imaginary part.
Try solving the equation x² = a. If a is positive, it's solvable within the real numbers — but the moment a becomes negative, there's no real answer.
Keep multiplying by i, and you cycle forever through just four values: 1, i, −1, −i. No matter how large n gets, its remainder when divided by 4 tells you the answer right away.
Adding complex numbers works like adding vectors; multiplying them multiplies the sizes and adds the angles. Pick two complex numbers and an operation, and check it on the complex plane.
A complex number can be written not just as "real part, imaginary part," but also as "distance from the origin (magnitude) and angle." This is called polar form, and points on the unit circle (radius 1) connect especially neatly to Euler's formula.
Electrical engineering uses complex numbers to handle an alternating current's size and phase (timing offset) at the same time. Circuit calculations mixing resistors, coils, and capacitors become far simpler thanks to complex numbers.
Signal and image processing relies on complex numbers at the core of the Fourier transform, which breaks sound or images apart into many waves. A music app's equalizer and photo compression both put this principle to work.
Quantum mechanics describes a particle's state with a wave function that is itself complex-valued. Without imaginary numbers, you couldn't even write down the equations of quantum mechanics in the first place.