What is a complex number, and why draw it as an arrow?
Ordinary numbers sit on a line: positive to the right, negative to the left. Complex numbers live in a plane. Each one is an arrow starting at the centre, described by two things: its length and its direction, the angle it makes with the line pointing right.
The same arrow can also be described by how far it goes right and how far it goes up. These two parts are called the real part and the imaginary part, and the arrow is written a + bi: a steps right, b steps up. The word "imaginary" is a historical accident. There is nothing imaginary about these numbers; they are simply arrows.
One complex number deserves a name of its own: i, an arrow of length 1 pointing straight up (90°). As the next lessons show, multiplying by i turns an arrow by 90°. Do it twice and an arrow pointing right ends up pointing left: i × i = −1. That is why i is sometimes called "the square root of −1". As an arrow, it is just a quarter turn.
An arrow of length 2 points straight up, at 90°. Which complex number is it? It goes 0 steps right and 2 steps up: 0 + 2i, that is 2i.
Why does physics need arrows? Because a wave has two properties at every moment: how big it is (its amplitude) and where it is in its cycle, rising or falling (its phase). An arrow holds exactly that: its length is the amplitude, its angle is the phase. As the wave oscillates, the arrow turns like the hand of a clock. Richard Feynman explained the whole theory of light this way in his book QED.
z = a + bi. Its length (modulus) is |z| = √(a^2 + b^2); its angle θ comes from tan θ = b / a.
The other way round: a = |z| cos θ and b = |z| sin θ, so z = |z| (cos θ + i sin θ).
Euler's formula writes this as z = |z| e^iθ. You do not need to know why yet: read e^iθ as "an arrow of length 1 at angle θ".
A complex number is an arrow with a length and an angle: perfect for a wave, whose amplitude is the length and whose phase is the angle.