Foundations — Exponential form z = re^(iθ)

Everything on the parent page rests on a handful of small ideas. Below we rebuild each one from nothing, in an order where every new symbol only uses symbols already explained. Nothing is assumed.

0. Two number lines make a plane

The picture below is the stage on which the entire topic happens. Look at the amber dot: to reach it you walk $x$ steps right, then $y$ steps up.

Why do we need a plane and not just a line? Ordinary numbers live on one line. Complex numbers carry two pieces of information at once, so they need a second direction — hence a whole flat sheet.

1. The symbol $i$ — the "quarter turn"

That algebra rule has a picture: multiplying by $i$ is a 90° anticlockwise turn on the plane.

• Start at $1$ (one step right). Multiply by $i$: you land at $i$ (one step up) — a quarter turn.
• Multiply by $i$ again: $i\cdot i=i^2=-1$ (one step left) — another quarter turn, now half-way round.
• Again: $i^3=-i$ (down). Again: $i^4=1$ (back to start).

Why does the topic need $i$? Because "multiply-to-rotate" is the seed of the whole idea: if multiplying by one special number turns you 90°, then multiplying by a general complex number will turn you by its angle. That is the punchline $\arg (z_1{z}_2)={\theta }_1+{\theta }_2$.

2. Modulus $r$ — the length of the arrow

Why this exact formula? Because $x$ and $y$ are the two sides of a right-angled triangle whose slanted side (the hypotenuse) is the arrow itself. Pythagoras says the hypotenuse squared equals the two shorter sides squared added — that is literally $r^2=x^2+y^2$.

3. Argument $\theta$ — the direction of the arrow

To connect the angle to the sides $x,y$ we need two trigonometric ratios. On the same right triangle:

• $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{x}{r}$, so $x=r\cos \theta$.
• $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}=\frac{y}{r}$, so $y=r\sin \theta$.

Why does the topic need $\theta$? Because "direction" is the second of the two facts an arrow carries, and turning is the operation that makes complex multiplication special.

4. The tangent and $\arctan$ — recovering the angle from $x,y$

If you know $x$ and $y$ but not $\theta$, how do you find the angle? Divide the two shadows: $\frac{y}{x}=\frac{r\sin \theta }{r\cos \theta }=\tan \theta$

Why introduce tan at all? Because it is the one tool that converts a ratio of shadows back into an angle — the reverse trip from $x,y$ to $\theta$.

5. The exponential $e^{(\cdot )}$ and factorials

Why does the topic need this series? Because it is the bridge from real exponentials to imaginary ones. The parent feeds $u=i\theta$ into this exact sum, and the cycling powers of $i$ (from §1) sort the terms into $\cos \theta$ and $\sin \theta$ — producing Euler's formula. Without the series there is no honest way to say what $e^{i\theta }$ even means.

6. Putting the symbols together

Now every symbol on the parent page is earned:

Read it left to right: a point → its right/up coordinates → its length-and-shadows → its compact length-and-turn form. Same arrow, four costumes.

Prerequisite map

Feed forward into the parent: the exponential form, and onward to Modulus and argument of a complex number, Polar form (trigonometric) r(cosθ + i sinθ), Complex multiplication as rotation and scaling, Taylor and Maclaurin series, De Moivre's theorem, Roots of unity and Euler's identity e^(iπ)+1=0.

Equipment checklist

Cover the right side and test yourself.