Complex polar form: what it means and how to use it
Represent a complex number by magnitude and direction.
Complex polar form is one of 3 complex numbers formulas in the precalculus section of this library, and it is used at high school · university level.
Why complex polar form works
Any point in the plane can be reached by choosing a distance and a direction instead of an across-and-up pair. Setting x = r cos θ and y = r sin θ converts one description into the other, and Euler's formula packages that pair into the single exponential e^(iθ), which is why the compact form is legitimate rather than just notation.
What each symbol means
$z=x+iy$, $r$ is modulus, and $\theta$ an argument.
Complex polar form: when it holds
For $z\ne0$, arguments differ by multiples of $2\pi$; the zero argument is undefined.
When it stops applying
The number 0 has modulus 0 but no argument at all, because a point sitting on the origin points nowhere. Every other complex number has infinitely many valid arguments, differing by full turns of 2π, so a principal value has to be chosen by convention.
Complex polar form: a worked example
$1+i=\sqrt2e^{i\pi/4}$.
The mistake to avoid
What people do: Students find the angle with the inverse tangent of y over x and accept whatever the calculator returns.
Why it goes wrong: Inverse tangent only reports angles between −90 and 90 degrees, so it cannot tell quadrant III from quadrant I. For −1 − i it returns 45 degrees, when the point actually sits at 225 degrees.
Do this instead: Sketch the point first, then adjust the calculator's answer by 180 degrees when the real part is negative. For −1 − i the correct argument is 225 degrees, or equivalently −135 degrees.
Complex polar form: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $z=x+iy$, $r$ is modulus, and $\theta$ an argument.
- Check the conditions before substituting. For $z\ne0$, arguments differ by multiples of $2\pi$; the zero argument is undefined.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most precalculus slips.
Where this formula fits
- Subject
- Precalculus formulas — 10 entries in this library
- Topic
- Complex numbers
- Level
- High school · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where complex polar form comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Quadratic Formula Explained lesson — the lesson behind this formula: use the formula and interpret its discriminant.
- Precalculus Calculator — check your substitution and the value it produces.
- Study precalculus — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 10 precalculus formulas — the full grouped reference, or the complete formula library.
Questions about complex polar form
Do I use degrees or radians in the exponential form?
Radians, always. The identity e^(iθ) = cos θ + i sin θ is built on radian measure, and feeding degrees into an exponential gives a number with no relation to the point you meant.
How do I convert 1 + i into polar form?
The modulus is √2, about 1.414, and the point sits at 45 degrees, or π/4 radians. So the polar form is √2 times e raised to iπ/4.
Why is polar form worth the trouble?
Because multiplication and powers become simple. Multiplying means multiplying the moduli and adding the arguments, which beats expanding brackets full of i terms every time.
Is the argument unique?
No, adding any whole number of full turns gives the same point. Most work uses the principal argument, chosen from the range −π to π, so answers can be compared reliably.