Math101 · Independent Ontario learning library
Math101Complex Numbers in Polar Form
Polar form represents a nonzero complex number by modulus and argument: $z=r(\cos\theta+i\sin\theta)$, often written $r\operatorname{cis}\theta$ or $re^{i\theta}$.
Polar form turns complex multiplication, division, powers, and roots into scaling and angle arithmetic, supporting waves, circuits, and rotations.
Intuition and core definition
Polar form represents a nonzero complex number by modulus and argument: $z=r(\cos\theta+i\sin\theta)$, often written $r\operatorname{cis}\theta$ or $re^{i\theta}$. Here $r=|z|>0$ is distance from the origin and $\theta$ is a directed angle from the positive real axis.
Notation, language, and conditions
For $z=a+bi$, $r=\sqrt{a^2+b^2}$. When $a\ne0$, $\tan\theta=b/a$ gives a reference value but still needs quadrant correction; using $\operatorname{atan2}(b,a)$ handles all quadrants and the case $a=0$. Arguments differ by $2\pi k$. Multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments.
Why this idea matters
Polar form separates a complex number's magnitude from its angle, making multiplication, division, and powers geometric operations.
A dependable method
- Plot $(a,b)$ or determine its quadrant.
- Compute modulus $r=\sqrt{a^2+b^2}$.
- Find a reference angle and choose an argument consistent with the quadrant.
- Write $r(\cos\theta+i\sin\theta)$ with angle units stated.
- Convert back using $a=r\cos\theta,b=r\sin\theta$ to verify.
