Each family of numbers appears the moment an operation breaks the previous set's closure: subtraction forces negatives (the integers ℤ), division forces fractions (the rationals ℚ), roots like √2 force the irrationals, and √−1 forces the imaginaries. Wrap them all together and you get the complex numbers, a + bi — which is what lets a Fourier transform recover phase as well as magnitude.
We’re not going to discuss the Fourier transform just yet — we’re going to discuss something without which it would be useless: numbers. What puzzles people about numbers is their very nature. What kinds are there? How are they derived? And what we’re leading to is the ultimate type — the complex number. You can perform a Fourier transform without complex numbers, but you’d only recover magnitude or power. To use the result fully — to recover phase, which matters just as much in filtering and image processing — you need complex numbers. In quantum mechanics they’re not a convenience but a necessity, laying the very foundations of physics.
The natural numbers: ℕ
Start with the counting numbers — the ones we use to buy lunch. Mathematically these are the natural numbers, ℕ = {0, 1, 2, …}. Are they closed under simple operations? Add any two and you get another natural number, so ℕ is closed under addition. Multiplication is just repeated addition, so it’s closed too. But subtraction breaks things: 10 − 20 = −10. We can’t have −2 cupcakes, yet we’ve produced a number that isn’t natural.
The integers: ℤ
To cater for negatives we extend ℕ with its mirror image to form the integers, ℤ = {…, −2, −1, 0, 1, 2, …} (the Z is from the German Zahlen). ℤ is a superset of ℕ and is closed under addition, multiplication and subtraction. But division breaks it: 5 ⁄ 10 = ½, which isn’t an integer. Time to extend again.
Rationals, irrationals and the reals
Numbers formed by dividing one integer by another are the rationals, ℚ — closed under addition, multiplication, subtraction and division. But roots break ℚ: √2 cannot be written as one integer over another (a classic proof by infinite descent). Its decimal runs on forever without repeating, so it’s irrational. Inside the irrationals sit the transcendentals — numbers like π and e that aren’t roots of any non-zero polynomial with rational coefficients. Astonishingly, almost all real and complex numbers are transcendental. Gather the rationals and irrationals together and you have the real numbers, ℝ.
Imaginary and complex numbers
The reals survive powers, but roots break them again: what is √−1? We define it as i, where i² = −1, so √−25 = 5i. Imaginary numbers turn previously unsolvable equations into solvable ones, just as negatives once unlocked x + 5 = 0. Group the reals and imaginaries together and you reach the final, all-encompassing class — the complex numbers, written:
Every number in mathematics is a complex number. When you pay €1 for something you’re really using a complex number with b = 0 — the imaginary part simply zeroed out. A few facts fall straight out: an integer can’t be transcendental; an imaginary number isn’t real (which rather makes sense); and zero is considered both real and imaginary.
You can also think of complex numbers as two-dimensional numbers that extend the one-dimensional number line out into the plane — a fascinating way to picture them.
Why this matters for energy
Complex numbers are exactly what let us analyse AC waveforms — magnitude and phase — which is the heart of interpreting electrical signals. It’s the foundation under work like energy-usage related signals and calculating usage from instantaneous data.