Two ways to name the same point
In the previous guide you learned to see a complex number z = x + i y as a point in the complex plane — go x to the right, then y up. That is the rectangular way of naming it: two distances along two perpendicular axes. But there is a second, equally honest way to point at the very same dot: say how far it sits from the origin, and in what direction. Those two numbers are the modulus |z| (the distance) and the argument arg z (the angle measured counterclockwise from the positive real axis).
Simple right-triangle trigonometry ties the two namings together. Dropping a perpendicular from z to the real axis gives a triangle with horizontal leg x, vertical leg y, and hypotenuse r = |z|. So x = r cos theta and y = r sin theta, where theta = arg z. Reading it the other way, r = sqrt(x^2 + y^2) and the angle satisfies tan theta = y / x. The point and its two descriptions never disagree; we are just choosing which pair of numbers to write down.
Substituting back gives the polar form of z: z = r (cos theta + i sin theta). This is the same number you started with, dressed in distance-and-direction clothing. The rectangular form is best for adding (line up the real parts, line up the imaginary parts), but as you are about to see, the polar form is where multiplication suddenly becomes simple.
Euler's formula: the bridge to the exponential
Now comes the single most useful identity in the whole subject. Euler's formula says that e^(i theta) = cos theta + i sin theta. The right-hand side is exactly the bracket sitting in the polar form, so we can compress everything into z = r e^(i theta). Distance becomes the size r in front; direction becomes the angle theta riding in the exponent. Where does it come from? Feed i theta into the exponential power series and sort the terms into the ones without an i and the ones with one: the i-free terms reassemble the cosine series, the i-carrying terms reassemble the sine series. Euler's formula is that reassembly written in one line.
Set r = 1 and the picture is sharp: e^(i theta) is the point on the unit circle at angle theta. Walk theta all the way round to 2 pi and you return to where you started, which already hints that the argument is only defined up to whole turns. Push theta to pi and you land at -1, giving the famous e^(i pi) + 1 = 0 — five constants, one true sentence.
Multiplication becomes turn-and-stretch
Here is the payoff. Take two numbers in polar form, z = r e^(i alpha) and w = s e^(i beta). Multiplying them, the sizes multiply and the angles ADD — because exponents add when you multiply powers: z w = r s e^(i (alpha + beta)). In plain words, multiplying by w rotates by its angle and scales by its size. That is the whole content of the rotation-and-scaling map: every complex number is secretly an instruction "turn by this much, stretch by this much."
z = r e^(i a), w = s e^(i b) z w = (r s) e^(i (a + b)) sizes multiply, angles add z/w = (r/s) e^(i (a - b)) sizes divide, angles subtract
Two tiny cases make this vivid. Multiplying by i = e^(i pi/2) is a quarter-turn counterclockwise with no change in size: that is exactly why i^2 = -1 (two quarter-turns is a half-turn, which sends 1 to -1). And multiplying by a positive real number like 3 = 3 e^(i 0) is a pure stretch with no turn. Every other complex number blends these two moves. Notice too that division just runs the angle clock backward: z / w divides the sizes and subtracts the angles.
De Moivre: powers are just repeated turns
If multiplying by z e^(i theta) turns by theta, then multiplying by it again turns by another theta, and so on. Raising z = r e^(i theta) to the n-th power therefore multiplies the size n times over and adds the angle to itself n times: z^n = r^n e^(i n theta). Written with cosines and sines this is de Moivre's theorem: (cos theta + i sin theta)^n = cos(n theta) + i sin(n theta). What would be a fearsome binomial expansion in rectangular form is, in polar form, a single multiplication of the angle.
Run a small example. To find a cube of (1 + i): its modulus is sqrt(2) and its argument is pi/4, so it is sqrt(2) e^(i pi/4). Cubing, the size becomes (sqrt(2))^3 = 2 sqrt(2) and the angle becomes 3 pi/4. That point is 2 sqrt(2) at angle 3 pi/4, which back in rectangular form is -2 + 2 i. No binomial slog, just "multiply the size three times, add the angle three times."
- Write z in polar form: read off the modulus r = |z| and an argument theta = arg z.
- Raise the size: the new modulus is r^n.
- Multiply the angle: the new argument is n theta (reduce by whole turns of 2 pi if you want it in the standard range).
- Convert back if needed: z^n = r^n (cos(n theta) + i sin(n theta)) = r^n cos(n theta) + i r^n sin(n theta).
Running it backward, and the angle's quiet subtlety
De Moivre also works in reverse, and this is where the next guide begins. If raising to the n-th power multiplies the angle by n, then taking an n-th root must DIVIDE the angle by n — and because the argument is only fixed up to whole turns, there are n different starting angles that all land on the right place after multiplying by n. That is exactly why an n-th root of a complex number has n distinct values, and why the roots of unity sit evenly spaced around the unit circle like the corners of a regular polygon. Guide 5 unfolds this fully.
Be careful with one honest pitfall: the angle is genuinely multivalued, and the equation tan theta = y / x cannot tell you which quadrant you are in. The number 1 + i and the number -1 - i give the same ratio y / x = 1, yet they sit on opposite sides of the plane. To pin the argument down you must look at the SIGNS of x and y, not just their ratio. This same multivaluedness, taken to functions, is the seed of the complex logarithm and branch cuts you meet much later — for now, just respect that arg z is a family of angles, not a single number.
Two more sanity checks worth carrying with you. First, the conjugate is a reflection across the real axis, which in polar terms just flips the sign of the angle: if z = r e^(i theta), then z-bar = r e^(-i theta). Second, e^(i theta) never has modulus other than 1, so it can never be zero — the exponential is never zero anywhere in the plane. Hold onto the central image: distance lives out front, direction lives in the exponent, and multiplication simply turns and stretches.