Complex Numbers — Formula & Trap Sheet

Every notation, formula, and trap across all 4 modules, on one page. Scan this the night before — if something here doesn't click, go back to that module's mastery check.

⚠️ The #1 trap in this chapter

$\sqrt{a}\times\sqrt{b}=\sqrt{ab}$ does NOT hold when both $a,b$ are negative. Treating $\sqrt{-1}\times\sqrt{-1}$ as $\sqrt{1}=1$ gives a contradiction — it's actually $i\times i=i^2=-1$. Always convert each $\sqrt{-a}$ to $\sqrt{a}\,i$ before multiplying, never after.

Module 1

Complex Numbers

$i=\sqrt{-1}$ $z=a+ib$ $\text{Re}(z)$ $\text{Im}(z)$

$z_1=z_2 \iff a=c$ AND $b=d$ — one complex equation hides two real ones

Every real number is also a complex number (with $b=0$) — reals are a special case, not a separate universe.
Equality of $a+ib=c+id$ always splits into TWO real equations — solve them separately.

Module 2

Algebra of Complex Numbers

$z_1+z_2$ $z_1z_2$ $i^{4k+r}$ $\sqrt{-a}=\sqrt{a}\,i$

$i^1=i,\ i^2=-1,\ i^3=-i,\ i^4=1$, then it repeats every 4 powers

To simplify $i^n$ for a big exponent: divide $n$ by 4, the remainder tells you the answer.
$\sqrt{-a}=\sqrt{a}\,i$ for positive $a$ — never write $\sqrt{-a}=-\sqrt{a}$, that's a completely different (wrong) thing.

Module 3

Modulus and Conjugate

$|z|=\sqrt{a^2+b^2}$ $\bar z=a-ib$ $z\bar z=|z|^2$ $z^{-1}=\frac{\bar z}{|z|^2}$

$z\bar z=|z|^2$ is always a real number — that's exactly why the conjugate trick clears $i$ from a denominator

$|z_1z_2|=|z_1||z_2|$ and $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$ — modulus behaves like ordinary absolute value.
To divide by a complex number: multiply top and bottom by the conjugate of the denominator.

Module 4

Argand Plane

real axis = x imaginary axis = y $z=x+iy \leftrightarrow P(x,y)$

$|z|=\sqrt{x^2+y^2}$ — the straight-line distance from the origin to $P(x,y)$, plain Pythagoras

The conjugate $\bar z$ is the mirror image of $P$ across the real axis — it only flips the y-coordinate's sign, never moves left/right.
Modulus is ALWAYS non-negative — it's a distance, and distances can't be negative.