Try solving the equation x² = −1. No real number works. Square a positive number and you get a positive number; square a negative number and you still get a positive number; square 0 and you get 0 — there's simply no real number in the world whose square is negative. But mathematicians didn't give up there. They imagined: "What if such a number did exist?" The number defined this way is i — we simply declare that i² = −1.
A real number multiplied by i is called an imaginary number, and a number of the form a+bi, adding a real number and an imaginary number, is called a complex number. A complex number can be drawn as a point on a plane whose horizontal axis is the real numbers and whose vertical axis is the imaginary numbers (the complex plane), which lets you treat a number as a location in a two-dimensional plane. Despite the name "imaginary," complex numbers form a complete number system with rules every bit as rigorous as the real numbers.
i's true charm shows up in its powers. i¹=i, i²=−1, i³=−i, i⁴=1, and starting from i⁵ it repeats the exact same cycle: i, −1, −i, 1. So whether it's i to the 100th power or the 1000th, you never have to multiply it all out — just find the remainder when the exponent is divided by 4, and you know the answer instantly. For example, i²³ has a remainder of 3 when 23 is divided by 4, so it equals i³, which is −i. This cycle through four values, going round and round, is the first hint that complex numbers are deeply connected to rotation.
You can do the four basic arithmetic operations with complex numbers too. Addition just means adding the real parts together and the imaginary parts together, which is exactly the same as adding two points like vectors on the complex plane. Multiplication, though, is a bit more interesting: when you multiply two complex numbers, their magnitudes multiply and their angles add. So multiplying a complex number by i means leaving its magnitude unchanged while rotating it by exactly 90 degrees. This is actually the real reason i²=−1 — multiplying by i twice adds 90 degrees twice, giving 180 degrees, which flips the direction to exactly the opposite way.
Representing a complex number this way, by its "magnitude and angle," is called polar form. In particular, complex numbers with magnitude 1 lie on the unit circle — a circle of radius 1 centered at the origin — and these points match exactly with Euler's formula, e^(iθ) = cosθ + i·sinθ. Plug in θ = π (180°) and you get e^(iπ) = −1, or in other words, e^(iπ) + 1 = 0. Five of the most important constants in all of mathematics — e, i, π, 1, and 0 — sit together in a single equation, which is why it's often called "the most beautiful formula in the world."
Because of its name, "imaginary" numbers can feel like a useless fantasy, but they're actually enormously practical. Electrical engineering uses complex numbers to calculate an alternating current's magnitude and phase at once, and complex numbers are also central to the Fourier transform used to break down sound and images. Even in quantum mechanics, the wave function that represents a particle's state is itself complex — you couldn't even write the equations without imaginary numbers. A number once thought "impossible to exist" is now used everywhere in practice, from smartphone signal processing to cutting-edge physics.
On our activity page, you can use a slider to change the value of a in x²=a and see how the solutions split into real and imaginary, watch the powers of i cycle through their four values, and pick two complex numbers to add and multiply, watching exactly what happens on the complex plane. At the end, you can even turn an angle slider yourself to trace out the unit circle drawn by Euler's formula.