1.13 Polar and Euler form

The Cartesian form $a + b\mathrm{i}$ describes a complex number by how far it is across and how far up on an Argand diagram. A point can equally be described by its distance from the origin and its direction, in the same way that a ship's position can be given by a distance and a bearing from a port. This second description, by modulus and argument, is the polar form. It makes multiplication and division much simpler than in Cartesian form, it shows what multiplication does geometrically, and through Euler's formula it links complex numbers with the exponential function and the circular functions.

Modulus and argument

For a non-zero complex number $z = a + b\mathrm{i}$, the modulus $r = |z| = \sqrt{a^2 + b^2}$

These relations come from the definitions of cosine and sine on a circle of radius $r$, and they hold in every quadrant, because $\cos\theta$ and $\sin\theta$ take the correct signs automatically.

The principal argument

Turning through a full turn of $2\pi$ brings a line back to the same direction, so $\theta$, $\theta + 2\pi$, $\theta - 2\pi$ and in general $\theta + 2k\pi$

Numbers above the real axis have positive principal arguments, numbers below it have negative ones, and a negative real number has argument $\pi$. The number $0$ has no argument.

Finding the argument in each quadrant

The equation $\tan\theta = \frac{b}{a}$ is true for every $z$, but it does not fix $\theta$: the calculator value $\arctan\frac{b}{a}$

For $z = -1 + \sqrt{3}\,\mathrm{i}$ and $w = -1 - \sqrt{3}\,\mathrm{i}$

Polar form

Substituting $a = r\cos\theta$ and $b = r\sin\theta$ into $z = a + b\mathrm{i}$

The number $\operatorname{cis}\theta$ itself has modulus $\sqrt{\cos^2\theta + \sin^2\theta} = 1$

Converting between Cartesian and polar form

To convert from Cartesian to polar form, find $r$ and $\theta$ as in the previous section. To convert from polar to Cartesian form, evaluate $r\cos\theta$ and $r\sin\theta$.

Recognising polar form

An expression is in polar form only if it matches $r(\cos\theta + \mathrm{i}\sin\theta)$ exactly: a positive real number $r$ outside the bracket, cosine first, a plus sign, and the same angle in both functions. Expressions that look similar can be rewritten using the symmetry identities $\cos(-\theta) = \cos\theta$

Euler form

The polar form can be written more compactly with the exponential function. The key fact, discovered by Leonhard Euler, links $\mathrm{e}$, $\mathrm{i}$ and the circular functions.

So $r\operatorname{cis}\theta$ and $r\mathrm{e}^{\mathrm{i}\theta}$ are two notations for the same number, and both appear in the formula booklet: $z = r(\cos\theta + \mathrm{i}\sin\theta) = r\mathrm{e}^{\mathrm{i}\theta} = r\operatorname{cis}\theta$

Why Euler's formula is reasonable

A full proof uses the power series of $\mathrm{e}^x$, $\sin x$ and $\cos x$, met in 5.19. Two simpler observations show why the formula fits.

First, the function $\operatorname{cis}\theta$ obeys the same law as powers. The product rule for polar form, proved with the compound angle identities, states that $\operatorname{cis}A \times \operatorname{cis}B = \operatorname{cis}(A + B)$, which is exactly the index law $\mathrm{e}^{\mathrm{i} A} \times \mathrm{e}^{\mathrm{i} B} = \mathrm{e}^{\mathrm{i}(A + B)}$

Second, differentiating $\cos\theta + \mathrm{i}\sin\theta$ with respect to $\theta$, treating $\mathrm{i}$ as a constant, gives

so $\operatorname{cis}\theta$ satisfies $\frac{d}{d\theta}f(\theta) = \mathrm{i} f(\theta)$, with $f(0) = 1$

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Euler's identity

Putting $\theta = \pi$ in Euler's formula gives $\mathrm{e}^{\mathrm{i}\pi} = \cos\pi + \mathrm{i}\sin\pi = -1$

This single equation connects five fundamental constants: $0$, $1$, $\pi$, $\mathrm{e}$ and $\mathrm{i}$. Other values follow in the same way: $\mathrm{e}^{\mathrm{i}\pi/2} = \mathrm{i}$

The three forms each have their uses. Cartesian form suits addition and subtraction; polar and Euler forms suit multiplication, division and powers.

Multiplying and dividing in polar form

The product rule

Multiplying two numbers in polar form, then using the compound angle identities from 3.10, gives a simple rule. With $z_1 = r_1\operatorname{cis}\theta_1$ and $z_2 = r_2\operatorname{cis}\theta_2$

In Euler form the same result is the index law: $r_1\mathrm{e}^{\mathrm{i}\theta_1} \times r_2\mathrm{e}^{\mathrm{i}\theta_2} = r_1 r_2\,\mathrm{e}^{\mathrm{i}(\theta_1 + \theta_2)}$

The quotient rule follows from the product rule. The reciprocal of $r\operatorname{cis}\theta$ is $\frac{1}{r}\operatorname{cis}(-\theta)$, because their product is $\operatorname{cis}0 = 1$

The rules make products of numbers given in Euler form a one-line calculation. Mixed questions often give the numbers in Cartesian form, so the first step is to convert.

Conjugates and reciprocals in polar form

The conjugate reflects $z$ in the real axis, which keeps the modulus and changes the sign of the argument. The reciprocal follows from the quotient rule with $z_1 = 1 = \operatorname{cis}0$. For $z = r\operatorname{cis}\theta = r\mathrm{e}^{\mathrm{i}\theta}$

So $z^*$ and $\frac{1}{z}$ point in the same direction; they differ only in length. In particular, $zz^* = r^2\mathrm{e}^{0} = r^2 = |z|^2$

Multiplication as enlargement and rotation

The product rule has a clear geometric meaning. Multiplying $z$ by $w = r\operatorname{cis}\alpha$ multiplies the distance from the origin by $r$ and adds $\alpha$ to the direction. On an Argand diagram, multiplication by $w$

Several special cases are worth knowing:

• Multiplying by $\mathrm{i} = \operatorname{cis}\frac{\pi}{2}$ is a rotation through $\frac{\pi}{2}$

To rotate a point $z$ through an angle $\alpha$ about a point $c$ instead of the origin, translate $c$ to the origin, rotate, and translate back: the image is $c + (z - c)\mathrm{e}^{\mathrm{i}\alpha}$

Rotations through a right angle need no trigonometry at all, since they are multiplications by $\pm\mathrm{i}$. This makes squares and right angles on an Argand diagram straightforward.

Sums in polar form

There is no simple rule for adding numbers in polar form: in general the sum must be found by converting to Cartesian form. One special case is worth knowing because it appears often. For $-\pi < \theta < \pi$, factorising out $\mathrm{e}^{\mathrm{i}\theta/2}$ gives

because $\mathrm{e}^{-\mathrm{i}\theta/2} + \mathrm{e}^{\mathrm{i}\theta/2} = 2\cos\frac{\theta}{2}$

Euler's formula and trigonometry

Replacing $\theta$ by $-\theta$ in Euler's formula gives $\mathrm{e}^{-\mathrm{i}\theta} = \cos\theta - \mathrm{i}\sin\theta$

These expressions turn trigonometric identities into algebra with exponentials, where the index laws do the work. They are an extension beyond the core content, but they give short proofs of results that are long with trigonometry alone.

The same approach, combined with De Moivre's theorem in 1.14, produces multiple angle formulae such as $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$.

Exam-style questions on polar and Euler form

Questions on this content often combine forms: a calculation is done twice, once in Cartesian form and once in polar form, and comparing the two results gives an exact value or an identity.

Questions are also set with the numbers given only by their moduli and arguments. These test the rules directly, and a GDC is no help.

A parameter question gives the argument of an expression and asks for an unknown. Write the expression in Cartesian form, then use $\tan\theta = \frac{b}{a}$ together with the quadrant.

The final question combines a calculator calculation with a geometric interpretation.

Self-check

• To do:

  I can find the modulus and principal argument of any non-zero complex number, sketching it first to get the quadrant right.

• To do:

  I can write a complex number in polar form $r\operatorname{cis}\theta$ and Euler form $r\mathrm{e}^{\mathrm{i}\theta}$, and convert back to $a + b\mathrm{i}$