How to write a complex number in polar form

Rectangular form tells us the real and imaginary parts of a complex number. Polar form describes the same point by its distance from the origin and its angle. We’ll find those two values and use them to rewrite the number.

Find the modulus

For $z = a + b i$ , the modulus is the distance from the origin. The real and imaginary parts form the legs of a right triangle, so

r = | z | = \sqrt{a^{2} + b^{2}}

For $z = - 2 - i$ , the modulus is $\sqrt{(- 2)^{2} + (- 1)^{2}} = \sqrt{5}$ .

Choose an angle in the correct quadrant

An argument $θ$ satisfies $\cos θ = a / r$ and $\sin θ = b / r$ . When $a \neq 0$ , the equation $\tan θ = b / a$ helps find a reference angle, and the signs of $a$ and $b$ determine the quadrant. Angles differing by $2 π$ describe the same direction.

For $- 2 - i$ , both parts are negative, so an argument is $π + \arctan (1 / 2)$ . For a point on an axis, we can read the angle directly. The number zero has modulus zero and no defined argument.

Use the polar formula

z = r (\cos θ + i \sin θ)

For $z = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i$ , the modulus is $1$ and the point lies in quadrant IV. We can use $θ = - π / 4$ .

z = \cos (- \frac{π}{4}) + i \sin (- \frac{π}{4})

The point lies in quadrant IV on the unit circle. Its angle is −π/4, or equivalently 7π/4.

For $z = 6 i$ , the modulus is $6$ and the angle is $π / 2$ , so $z = 6 (\cos (π / 2) + i \sin (π / 2))$ .

Convert back or use exponential form

\begin{array}{l} a = r \cos θ \\ b = r \sin θ \end{array}

Euler’s formula gives $e^{i θ} = \cos θ + i \sin θ$ , so we can also write $z = r e^{i θ}$ . All three forms describe the same complex number.

Learn mathKrista KingAugust 17, 2021math, learn online, online course, online math, equilibrium solutions

How to write a complex number in polar form

Find the modulus

Choose an angle in the correct quadrant

Use the polar formula

Convert back or use exponential form

Courses