MISSION · Number sense
Polar Form and De Moivre's Theorem
Multiplying complex numbers is stretching and turning
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The idea in one sentence
Every complex number is a distance and a direction from the origin, and writing it that way turns multiplication into “stretch by this much and turn by that much”.
Modulus and argument
On the Argand diagram, \(z = a + bi\) is the point \((a, b)\). Two numbers locate it just as well:
The modulus \(|z| = r = \sqrt{a^2 + b^2}\) — its distance from the origin, straight out of Pythagoras.
The argument \(\arg z = \theta\) — the angle from the positive real axis, measured anticlockwise.
For \(z = 3 + 4i\): \(r = \sqrt{9 + 16} = 5\) and \(\theta = \arctan\frac43 \approx 53.1°\).
Then \(a = r\cos\theta\) and \(b = r\sin\theta\), so
\[z = r(\cos\theta + i\sin\theta)\]which is the polar form. It is often abbreviated \(r\,\text{cis}\,\theta\), and in calculus it becomes Euler’s \(re^{i\theta}\) — the same thing written more usefully.
Watch the quadrant. \(\arctan\) only ever returns angles in two of the four, so for \(-3 + 4i\) the calculator’s \(-53.1°\) needs \(180°\) added to give the true \(126.9°\). Sketch the point before trusting the arctangent.
Why anyone bothers
In rectangular form, multiplication is four products and a collect. In polar form it is two operations, and both are geometric:
\[r_1(\cos\theta_1 + i\sin\theta_1) \cdot r_2(\cos\theta_2 + i\sin\theta_2) = r_1 r_2\big(\cos(\theta_1 + \theta_2) + i\sin(\theta_1+\theta_2)\big)\]Moduli multiply, arguments add. Division does the reverse: moduli divide, arguments subtract.
So multiplying by a complex number is a stretch combined with a rotation. Multiplying by \(i\) — modulus 1, argument \(90°\) — is a pure quarter turn, which is where the intuition that “\(i\) rotates” comes from, and why \(i^2 = -1\) stops looking strange: two quarter turns is a half turn, which is multiplying by \(-1\).
The addition formula for angles does all the work here. It is the trig identity \(\cos(A+B) = \cos A\cos B - \sin A \sin B\), appearing exactly where the \(i^2 = -1\) produces a minus sign.
De Moivre’s theorem
Multiply a number by itself \(n\) times and the rule compounds:
\[\big(\cos\theta + i\sin\theta\big)^n = \cos n\theta + i \sin n\theta\]Raising to a power multiplies the argument and raises the modulus to that power. So \((\cos 30° + i\sin 30°)^6 = \cos 180° + i \sin 180° = -1\).
Doing that expansion in rectangular form would take a page. This takes a line.
Roots, and why there are always n of them
De Moivre also runs backwards, and this is where it earns its place.
To find the cube roots of 8: write \(8\) as modulus 8, argument \(0°\). The roots have modulus \(\sqrt[3]{8} = 2\) and argument \(\frac{0°}{3} = 0°\) — but \(8\) also has arguments \(360°\) and \(720°\), since going round again lands in the same place. Dividing those by 3 gives \(120°\) and \(240°\).
\[2(\cos 0° + i\sin 0°) = 2, \qquad 2\,\text{cis}\,120°, \qquad 2\,\text{cis}\,240°\]Three cube roots, not one. In general a non-zero complex number has exactly \(n\) distinct \(n\)th roots, equally spaced around a circle of radius \(\sqrt[n]{r}\) — they sit at the vertices of a regular \(n\)-gon. The \(n\)th roots of 1, called the roots of unity, are the most-used case.
This is the concrete meaning of the fundamental theorem of algebra: \(z^n = c\) has exactly \(n\) solutions, and polar form shows you where they all are.
Common mix-ups
- Wrong quadrant from a bare \(\arctan\).
- Adding moduli. They multiply; the arguments add.
- Finding only one root. An \(n\)th root question has \(n\) answers unless it says otherwise.
- Mixing degrees and radians inside one calculation.
- Dropping the modulus in De Moivre. The full statement is \(\big(r\,\text{cis}\,\theta\big)^n = r^n \text{cis}\, n\theta\); the \(r^n\) is easy to lose.
Check yourself
Find the modulus of \(3 + 4i\), say what happens to the arguments when you multiply, and evaluate \((\cos 30° + i \sin 30°)^6\).
\(\sqrt{3^2 + 4^2} = 5\). Multiplying multiplies the moduli and adds the arguments. And De Moivre gives \(\cos 180° + i\sin 180°\), so \(\theta = 180°\) and the value is \(-1\).
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