Foundations — Polar form — r(cos θ + i sin θ) = r·cis θ

Before you can read the parent note Polar form fluently, every symbol in it must mean something pictured, not just written. This page builds each one from nothing, in the order they depend on each other.

1. The imaginary unit $i$ — a 90° turn

Why does the topic need it? Ordinary numbers live on a single line (left–right). To describe arrows that also go up and down, we need a second, perpendicular direction. That "perpendicular direction" is exactly what $i$ marks.

Look at the figure. Start with the arrow $1$ pointing right. Multiply by $i$ once → it swings to point up. Multiply again → it points left, and indeed $i\times i=i^2=-1$, which is the arrow pointing left. So "$i^2=-1$" is not a mystery: it is "turn 90°, then 90° more = face backwards".

2. The complex number $z=a+bi$ — an arrow's two coordinates

The picture: draw two number lines crossing at a point $O$ (the origin). The horizontal line measures $a$, the vertical line measures $b$. The number $z=a+bi$ is the single point you reach by going $a$ right and $b$ up — and the arrow from $O$ to that point. This plane is the Argand Diagram.

3. The modulus $r=|z|$ — the length of the arrow

Why this exact formula? Look at the figure. Drop a straight line from the tip of the arrow down to the horizontal axis. You get a right-angled triangle: the bottom leg has length $|a|$, the upright leg has length $|b|$, and the arrow itself is the slanted longest side (the hypotenuse). Pythagoras — "the square of the long side equals the sum of the squares of the two short sides" — gives $r^2=a^2+b^2$, so $r=\sqrt{a^2+b^2}$.

Why the tool "square root"? Because Pythagoras hands us $r^2$, and we want $r$ itself, we must undo the squaring — that is precisely what x​ does.

The full ingredient is developed in Modulus and Argument.

4. The angle $\theta$ and the argument $\arg z$ — the direction of the arrow

Why measured from the positive real axis, anticlockwise? It is a shared convention so everyone reads the same arrow the same way — like agreeing that a compass starts from North. Anticlockwise-positive matches how $i$ turns arrows (Section 1), so multiplication and angles stay consistent.

We measure angles in radians: a full turn is $2\pi$, a half turn is $\pi$, a quarter turn is $\frac{\pi }{2}$. Radians (not degrees) are used because every deeper tool — De Moivre's Theorem, Euler's Formula — is clean only in radians.

5. $\cos \theta$ and $\sin \theta$ — turning an angle back into coordinates

In plain words: $\cos \theta$ is "what fraction of the arrow's length points sideways", and $\sin \theta$ is "what fraction points upward". They are the dictionary between an angle and the sideways/upward reach of the arrow.

Rearranging (multiply both by $r$): $a=r\cos \theta ,b=r\sin \theta .$ This is the exact bridge the parent uses to turn $r$ and $\theta$ back into $a+bi$ — and it is the whole content of polar form.

6. $\tan \theta$ and ${\tan }^{-1}$ — recovering the angle from the coordinates

Why this tool? When converting Cartesian → polar we know $a$ and $b$ but not $\theta$. We need something that eats the ratio $b/a$ and gives back the angle — that is precisely ${\tan }^{-1}$.

7. The four quadrants — where the arrow lives

Why the topic needs this: because ${\tan }^{-1}$ alone cannot tell Q I from Q III, or Q II from Q IV, you must first see which quadrant $(a,b)$ sits in, then adjust — the exact table in the parent note. Every worked example there begins "which quadrant?" for this reason.

How these foundations feed the topic

Equipment checklist

Connections

• Argand Diagram — the plane these symbols live in.
• Modulus and Argument — the two ingredients $r$ and $\theta$ built here.
• Multiplication and Division of Complex Numbers — where these foundations pay off.
• De Moivre's Theorem — needs the radian angle $\theta$ from Section 4.
• Euler's Formula — the deeper reason $\cos \theta +i\sin \theta$ behaves like an exponent.
• Roots of Complex Numbers — reuses modulus and argument.