Foundations — Modulus - z - and argument arg(z)

Before you can read the parent note on Modulus and Argument, you need every single symbol it throws at you to already feel obvious. This page builds them one at a time, from nothing, each one earning its place before the next appears.

0. The plane we draw everything on

Everything below lives on one picture: a flat sheet with a horizontal line and a vertical line crossing at a centre point. This is the Argand Diagram — but for now just think of it as ordinary graph paper.

1. The symbols $x$ and $y$ — sideways and up

Notice both can be negative. If $x=-3$ you walk left; if $y=-4$ you walk down. Hold onto that — the signs of $x$ and $y$ are the whole reason the argument is tricky later.

2. The symbol $i$ — the "turn upward" marker

We do not need to fully understand why $i^2=-1$ here — that belongs to Complex Numbers - Cartesian form z = x + iy. We just need to know it exists and marks the vertical direction.

3. The symbol $z$ — a whole point in one letter

4. The symbols $x$ and $y$ as the sides of a right triangle

Here is the picture that unlocks everything. Drop the dot $z$ onto the plane, then draw:

• the horizontal leg of length $|x|$,
• the vertical leg of length $|y|$,
• the slanted arrow from origin to dot.

5. The square root symbol  ​ and Pythagoras

Squaring $x$ and $y$ also quietly kills the minus signs ($(-3{)}^2=9$), which is why a distance can never come out negative.

6. The angle $\theta$, and why we measure it anticlockwise

We measure angles in radians: a full circle is $2\pi$, a half-circle (pointing left) is $\pi$, a quarter-turn up is $\pi /2$. Radians are just a unit for turning, like using metres instead of feet.

7. The trio $\sin$, $\cos$, $\tan$ — how the angle sets the sides

On our right triangle, name the sides relative to the corner angle $\theta$:

• adjacent = the leg next to $\theta$ (the horizontal run $x$),
• opposite = the leg across from $\theta$ (the vertical rise $y$),
• hypotenuse = the arrow, length $r$.

8. The undo-button $\arctan$, and its blind spot

9. Quadrants — the four neighbourhoods of the plane

10. The bar $\bar{z}$ — the mirror-flip

How these feed the topic

Once you own the top row of that map, the parent note reads like a story you already know: the length comes from Pythagoras, the direction comes from the angle, and polar form just staples the two together.

Equipment checklist

Test yourself — cover the right side and answer out loud.

Ready? Head back to the parent topic and it will feel like every symbol was already an old friend. From here you can also branch into Polar and Trigonometric form of complex numbers and, later, Euler's formula e^{iθ} = cosθ + i sinθ.