Math101 · Independent Ontario learning library
Math101learn.math101.caComplex 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.
Worked example
Representations and interpretation
The complex plane treats $a+bi$ as a vector from the origin. Rectangular coordinates give horizontal/vertical components; polar coordinates give length/direction. Multiplication rotates and scales this vector.
Reasoning about variations
The number zero has modulus zero but no unique argument, so polar angle is undefined there. The same nonzero number has infinitely many forms with angles $\theta+2\pi k$.
Common mistakes
How to check your work
- Convert polar form back to rectangular components.
- Confirm modulus is nonnegative and matches point distance.
- Check the argument lands in the correct quadrant.
Practice
- Write $1+i$ in polar form using $0\le\theta<2\pi$.
- Multiply moduli $3$ and $4$ in polar-form multiplication.
- What are all arguments corresponding to $\pi/3$?
Answers and brief solutions
Show answers
- $\sqrt2(\cos\frac\pi4+i\sin\frac\pi4)$ $r=\sqrt2$ and the point lies at angle $\pi/4$.
- $12$ Complex moduli multiply.
- $\pi/3+2\pi k$, $k\in\mathbb Z$ Full rotations give the same direction.
Synthesis and transfer
Multiplying two alternating-current phasors adds their arguments and multiplies their magnitudes; converting back to rectangular form checks both quadrant and scale.
Let $z_1=2\operatorname{cis}(30^\circ)$ and $z_2=3\operatorname{cis}(100^\circ)$. Their product is $6\operatorname{cis}(130^\circ)$: lengths multiply while rotations add. Division would instead produce magnitude $2/3$ and argument $-70^\circ$, which can be rewritten coterminally as $290^\circ$. Converting the product to rectangular components checks the quadrant and approximate signs. The zero complex number needs separate treatment because its modulus is zero but its argument is not uniquely defined. This geometric arithmetic is especially efficient for repeated powers, where De Moivre's theorem raises the modulus and multiplies the angle.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Write $1+i$ in polar form using $0\le\theta<2\pi$.
- $r=\sqrt2$ and the point lies at angle $\pi/4$.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
