Complex plane

The complex plane (also the Argand diagram or Gaussian plane) draws the complex number $a + bi$ as the point $(a, b)$: the real part across, the imaginary part up. Its distance from 0 is the modulus $|z| = \sqrt{a^2 + b^2}$, and its direction is the argument $\arg z$, the angle from the positive real axis; the page gives its principal value, in $(-180^\circ, 180^\circ]$, and none at 0. So $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$, the polar form. The page shows what adding, multiplying and repeatedly multiplying by a number $w$ do to the whole plane.

Multiplying by $w$ sends 1 to $w$ and $i$ to $iw$, which is $w$ turned 90°. The faint unit grid is carried to the yellow grid spanned by $w$ and $iw$: the whole plane turns by $\arg w$ about 0 and stretches by $|w|$, so the image grid is always square. A point z = $a + bi$ keeps its grid address, so $zw = a\cdot w + b\cdot(iw)$ — Component breakdown draws both walks. With $w = c + di$, $iw = -d + ci$: turning $w$ a quarter turn moves its imaginary part $d$ onto the negative real axis, and that is the $-bd$ in the formula, the trace of $i^2 = -1$. The triangles 0–1–$z$ and 0–$w$–$zw$ are similar (for $w \ne 0$), which is why lengths multiply and angles add; a sum past 180° is brought back by 360°. Multiplying by 0 sends every point to 0. Dividing by $w \ne 0$ undoes it all: turn back by $\arg w$ and shrink by $|w|$.

Add slides the whole plane by $w$ (the parallelogram 0, $z$, $w$, $z + w$), and $|z + w| \le |z| + |w|$, with equality only when $z$ and $w$ point the same way. Repeat marks $z, zw, zw^2, \dots$: each step turns by $\arg w$ and stretches by $|w|$, so the points lie on an equiangular spiral. With $|w| = 1$ and $\arg w$ a fraction $p/q$ of a turn they come back after $q$ steps (from 1: the $q$th roots of unity); with $w = e^{i\cdot 1}$ they never do, since $1/2\pi$ is irrational.