Operations with Complex Numbers in Polar Form

Polar Form: A Quick Refresher

Every nonzero complex number $z = a + bi$ is a point $(a, b)$ in the complex plane. (Some problems call this point the affix of $z$. It is just the point that represents $z$.) You can describe that point by its distance from the origin and its direction.

To go between the two forms, use right-triangle trigonometry:

• Polar to Cartesian: $a = r\cos\theta$ and $b = r\sin\theta$.

• Cartesian to polar: $r = \sqrt{a^2 + b^2}$. Find the reference angle with $\tan^{-1}\left|\tfrac{b}{a}\right|$, then place $\theta$ in the correct quadrant.

Two quick facts about polar form are useful all through this page.

For example, if $z = 7_{110^\circ}$, then $\bar z = 7_{-110^\circ} = 7_{250^\circ}$ and $-z = 7_{290^\circ}$.

Multiplying and Dividing in Polar Form

Multiplying two complex numbers in Cartesian form takes four products. In polar form it is almost free.

Why? Multiply the two trigonometric forms and use the angle-sum formulas:

The real part is $\cos(\alpha+\beta)$ and the imaginary part is $\sin(\alpha+\beta)$. So the result is $rs\,[\cos(\alpha+\beta) + i\sin(\alpha+\beta)]$.

The figure shows $z_1 = 2_{30^\circ}$ times $z_2 = 5_{45^\circ}$. The product has modulus $2 \cdot 5 = 10$, which is 5 times as long as $z_1$, and argument $30^\circ + 45^\circ = 75^\circ$.

With several factors, do the moduli and the angles separately. For a chain such as $\dfrac{a_\alpha \cdot b_\beta}{c_\gamma}$ the result is $\left(\dfrac{ab}{c}\right)_{\alpha+\beta-\gamma}$.

Multiplying Is Rotating and Scaling

The product rule has a geometric meaning. Multiplying $z$ by $s_\beta$ does two things at once:

• It scales the distance from the origin by the factor $s$.

• It rotates the point $\beta$ degrees counterclockwise around the origin.

The Inverse of a Complex Number

Dividing by $z$ is multiplying by $\dfrac1z$. Put $1 = 1_{0^\circ}$ in the numerator of the quotient rule and you get the inverse.

A mixed problem: the total impedance of two parallel components in an AC circuit satisfies $\dfrac{1}{Z_T} = \dfrac{1}{Z_1} + \dfrac{1}{Z_2}$. Reciprocals are easy in polar form, but addition is not. You must switch to Cartesian form for the sum.

De Moivre's Theorem: Powers

A power is repeated multiplication. Apply the product rule $n$ times to $r_\theta$: the modulus is multiplied by itself $n$ times, giving $r^n$, and the argument is added to itself $n$ times, giving $n\theta$.

Now back to the opening question: $(1+i)^{20}$. We have $1 + i = (\sqrt2)_{45^\circ}$, so $(1+i)^{20} = \big((\sqrt2)^{20}\big)_{20\cdot 45^\circ} = 1024_{900^\circ}$. Since $900^\circ - 720^\circ = 180^\circ$, the result is $1024_{180^\circ} = -1024$. Twenty multiplications became one.

Roots of a Complex Number

The equation $x^2 = 4$ has two real solutions, $\pm 2$. Over the complex numbers, $z^n = w$ always has exactly $n$ solutions (this is the Fundamental Theorem of Algebra at work). Turn De Moivre's Theorem around to find them.

Why? If $z = r_\alpha$ and $z^n = s_\beta$, De Moivre gives $r^n = s$ and $n\alpha = \beta + 360^\circ k$ (two angles name the same direction if they differ by whole turns). Solve: $r = \sqrt[n]{s}$ and $\alpha = \dfrac{\beta + 360^\circ k}{n}$. Raising $k$ by one adds $\dfrac{360^\circ}{n}$. At $k = n$ you have added a full $360^\circ$ and the list starts repeating, so $k = 0, \ldots, n-1$ gives exactly $n$ different roots.

When the equation is not already of the form $z^n = w$, rearrange it first and then convert $w$ to polar form. For $z^4 + 81 = 0$, write $z^4 = -81$ and note that $-81 = 81_{180^\circ}$.

Roots as Regular Polygons

All $n$ roots of $z^n = w$ lie on a circle of radius $R = \sqrt[n]{|w|}$ and are spaced $\dfrac{360^\circ}{n}$ apart. So they are the vertices of a regular $n$-gon centered at the origin. That means you can use geometry on them.

Shifted Roots and Hidden Systems

Some problems look harder but reduce to what you already know.

When a problem says "the root with argument in the first quadrant" (or similar), compute all $n$ roots first and keep the one whose argument fits the condition. If a problem asks for "the transformation equivalent to multiplying by a root," pick that root and then multiply with the product rule: the rotation angle is the argument of that root.

Using De Moivre's Theorem as a Tool

De Moivre's Theorem connects algebra and trigonometry. Two techniques come up again and again.

Technique 1: equate the real and imaginary parts. Expand $(\cos\alpha + i\sin\alpha)^n$ in two ways: with De Moivre, and with the binomial expansion. Two complex numbers are equal when their real parts are equal and their imaginary parts are equal.

For the cube, expand with $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ with $b = i\sin\alpha$, and remember $i^2 = -1$, $i^3 = -i$, and $\sin^2\alpha = 1 - \cos^2\alpha$.

Technique 2: compare moduli, then arguments. When an equation mixes $z$ and $\bar z$, write $z = r_\theta$ and $\bar z = r_{-\theta}$, then set the moduli equal and the angles equal (up to full turns).

Technique 3: when does a power land on the real axis? The point $(r_\theta)^n$ is a positive real number when $n\theta$ is a multiple of $360^\circ$, and a negative real number when $n\theta = 180^\circ + 360^\circ k$. For example, $(1 + i)^n = \big((\sqrt2)^n\big)_{45^\circ n}$ is a positive real number when $45n$ is a multiple of $360$, so the smallest positive $n$ is $8$.

Summary