Key takeaway: The polar form of a complex number writes z = a + bi as r(cos θ + i sin θ), where r is the distance from the origin and θ is the angle from the positive real axis.
Grade 12 – PrecalculusN-CN.B.4Numbers and Operations
Definition of Polar Form of a Complex Number
The polar form of a complex number describes a point on the complex plane by distance and heading instead of by horizontal and vertical steps. The distance \(r\) is the modulus, and the heading \(\theta\) is the argument.
Rectangular form \(a+bi\) is best for adding and subtracting. Polar form wins for multiplying, dividing and raising to powers, because moduli multiply and angles add. Converting needs only the Pythagorean theorem for \(r\) and an inverse tangent for \(\theta\), with a check of the quadrant so the angle points toward the real location of the number.
Formula and notation
For \(z=a+bi\):
\[r=\sqrt{a^2+b^2},\quad \tan\theta=\frac{b}{a},\quad z=r(\cos\theta+i\sin\theta).\]
The compact form \(r\,\text{cis}\,\theta\) is common. Going back: \(a=r\cos\theta\) and \(b=r\sin\theta\).
Key facts
- In the polar form of a complex number, adding 360° (or 2π) to the angle gives the same point.
- To multiply, multiply the moduli and add the angles.
- A positive real number has angle 0; a negative real number has angle 180°.
- The modulus equals |z|, the distance from the origin to the point.
Polar Form of a Complex Number example
Write \(z=-3+3i\) in polar form.
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Distance from the origin: square both parts, add to get 18, then take the root. The modulus is \(3\sqrt{2}\), about 4.24.
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The two parts have equal size, so the slope ratio is 1 and the acute angle to the real axis is 45 degrees.
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A negative real part with a positive imaginary part puts the point in quadrant II, so subtract 45 from 180 to get 135 degrees.
In polar form, \(z=\)\(3\sqrt{2}\,(\cos 135^{\circ}+i\sin 135^{\circ})\).
Common mistake
Watch out
Pablo computes \(\theta=\tan^{-1}(\frac{3}{-3})=-45^{\circ}\) and writes \(z=3\sqrt{2}(\cos(-45^{\circ})+i\sin(-45^{\circ}))\).
Why it is wrong
The calculator’s inverse tangent only returns angles in quadrants I and IV. That polar form of a complex number would describe \(3-3i\). Locate the number first, see quadrant II, and add 180° to get 135°.
Related terms and lessons
- De Moivre's Theorem
- Polar Coordinates
- Magnitude of a Vector
- Unit Circle
- Fundamental Theorem of Algebra
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