The words "complex number" scare people off. They needn't. A complex number is just an arrow that is allowed to spin — a clock hand. Quantum amplitudes are these spinning arrows, and their turning is what produces interference, the engine of the whole subject.
↩ before you start · keep these handy
·An arrow has a length and a direction, and lives as two numbers [across, up] (page 0.1). Everything here is that same picture, renamed.
·You can multiply two ordinary numbers. That is the only arithmetic used below.
·No need to "believe in" imaginary numbers — by the end you will see they are nothing spookier than the up-direction.
🔑 symbol decoder · every new mark, in plain words
zthe name of a complex number — one arrow in a flat plane.a + b·ithat arrow written out: a across, b up. The i just tags which number is the "up" one.ithe operation "turn 90°." Not a size — a quarter-turn. (And i² = −1 is just "two quarter-turns = a half-turn = flip sign.")Re, Im"real part" (the across number) and "imaginary part" (the up number).|z|the length of the arrow, exactly as on page 0.1.phasethe angle the arrow points (also called its "argument"). Spinning the arrow changes only this.e^(iθ)shorthand for "a unit arrow turned to angle θ." Don't panic at e — here it only ever means "point this way."
feel
A number that also points somewhere
An ordinary number lives on a line: 3 is three steps right, −2 is two steps left. A complex number lifts off that line into a plane. It still has a size, but now also an angle — a direction it leans. The horizontal part is the familiar real number; the vertical part is labelled imaginary, an unfortunate name for nothing more than "the up direction."
🕐 everyday picture
Think of the second hand on a clock. It has a fixed length and it sweeps through every angle as it ticks. A complex number is exactly that hand frozen at one instant — a length and an angle. "Multiplying by i" is one tick of 90°; do it four times and you are back where you started, just as four quarter-turns make a full circle.
recapA complex number is just an arrow in a plane — a size and an angle, like a clock hand.
play
The phasor — a hand on a clock
Drag the tip around. The real part is its shadow on the floor, the imaginary part its shadow on the wall. Hit spin to watch it turn, and × i to see the one fact worth memorising: multiplying by i is a quarter‑turn.
▸ complex planez = a + b·i
z{{ zStr }}
real a{{ za }}
imag b{{ zb }}
size |z|{{ zmag }}
angle{{ zarg }}°
|z|² = a² + b² — the size squared. This becomes a probability on the next page.
recapSpinning changes the angle but not the size; multiplying by i is exactly one 90° tick.
see
Two arrows added: interference
Here is why phase matters. Take two arrows of the same size and add them tip‑to‑tail. When they point the same way they reinforce — a long result. When they point opposite ways they cancel to nothing. Slide the phase and watch the sum breathe between the two. This cancelling — two somethings making nothing — is impossible for plain probabilities, and it is the whole reason quantum mechanics is strange.
Euler's formula is just the dictionary between the two ways of describing the same arrow — angle‑and‑size, or across‑and‑up:
eiθ = cos θ + i·sin θ
a unit arrow at angle θ has real part cos θ and imaginary part sin θ.
multiplying by e^(iφ) adds φ to the angle — a pure rotation, no resizing.
worked example · read an arrow, then turn it
(a) how big and which way is z = 3 + 4i?
1.size: |z| = √(3² + 4²) = √25 = 5
2.angle: 3 across and 4 up → about 53° above the floor
(b) what does multiplying by i do to z = 2 + 1i?
3.i·(2 + i) = 2i + i² = 2i − 1 = −1 + 2i
4.the point (2, 1) moved to (−1, 2) — a clean quarter‑turn, same length. ✓
recape^(iθ) is just "a unit arrow pointed at angle θ"; multiplying by it rotates.
⚠ common misconceptions
"Imaginary numbers aren't real, so they can't describe anything physical." The name is just a 400-year-old insult that stuck. The "imaginary" axis is simply the up direction of the plane — every bit as real as the across direction. Nothing fake is going on.
"i must be some huge or tiny number." It has no size in that sense — i is an action: "rotate 90°." That is why doing it twice (i²) lands on −1: two quarter-turns make a half-turn, which flips a number's sign.
✓ you can now
✓read a complex number as an arrow with a size and an angle
✓see "multiply by i" as a 90° turn, and explain i² = −1 without any mystery
✓predict when two equal arrows reinforce or cancel — the seed of all interference