1.13 Polar and Euler form

Learning objectives

• Express a complex number in polar (modulus–argument) form $r(\cos\theta + \mathrm{i}\sin\theta) = r\,\mathrm{cis}\,\theta$ and in exponential form $r\mathrm{e}^{\mathrm{i}\theta}$.

• Convert between Cartesian, polar and exponential forms, by hand for exact values and with technology.

• Calculate products and quotients of complex numbers in polar or exponential form, giving the argument in the required range.

• Calculate integer powers of complex numbers in polar or exponential form.

• Describe addition and subtraction of complex numbers as vector addition and subtraction on an Argand diagram.

• Describe multiplication by a complex number as a rotation combined with a stretch about the origin.

• Find the sum of two sinusoidal functions with the same frequency, in the form $A\cos(Bt + C)$, using complex numbers.

• Interpret the amplitude and phase of a combined voltage or current in an AC circuit.

Introduction

A complex number can be located on an Argand diagram in two ways: by its horizontal and vertical coordinates, which give the Cartesian form $a + b\mathrm{i}$, or by its distance from the origin and its direction, which give the polar form. In polar form multiplication is simple, because lengths multiply and angles add. Euler's formula then rewrites polar form as a power of $\mathrm{e}$, so complex numbers obey the familiar laws of exponents.

Engineers use these ideas to analyse alternating current. Mains electricity supplies a voltage that rises and falls as a cosine wave 50 or 60 times a second, and engineers represent each such wave by a single complex number that records its size and timing. Adding two waves of the same frequency then becomes adding two complex numbers.

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Polar form

From modulus and argument to polar form

Subtopic 1.12, Complex numbers, described the point $z = a + b\mathrm{i}$ by its modulus $r = |z|$ and its argument $\theta = \arg z$

The definitions of sine and cosine on the unit circle extend this to every quadrant, so for any non-zero complex number

Writing $\cos\theta + \mathrm{i}\sin\theta$ in full every time is slow, so it has a short name.

The polar form of a number is not unique, because adding $2\pi$ to the argument describes the same direction: $2\,\mathrm{cis}\,\frac{\pi}{3}$ and $2\,\mathrm{cis}\,\frac{7\pi}{3}$

Converting from Cartesian to polar form

Converting to polar form means finding $r$ and $\theta$, exactly as in subtopic 1.12.

1. Plot the point roughly to see its quadrant.

2. Find $r = \sqrt{a^2 + b^2}$.

Converting from polar to Cartesian form

Converting back uses $a = r\cos\theta$ and $b = r\sin\theta$, or simply expands the bracket. Exact values of sine and cosine at multiples of $\frac{\pi}{6}$

Euler's formula and exponential form

Why cis θ behaves like a power

Multiplying two numbers of modulus 1 in polar form shows a pattern. Expanding $\mathrm{cis}\,\alpha \times \mathrm{cis}\,\beta$ and using the compound angle identities $\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$

So $\mathrm{cis}\,\alpha \times \mathrm{cis}\,\beta = \mathrm{cis}(\alpha + \beta)$: multiplying adds the angles. This is exactly how powers behave, since $x^{\alpha} \times x^{\beta} = x^{\alpha + \beta}$

Euler's formula works only with $\theta$ in radians. In the derivative argument, the rules $\frac{\mathrm{d}}{\mathrm{d}\theta}\sin\theta = \cos\theta$ and $\frac{\mathrm{d}}{\mathrm{d}\theta}\cos\theta = -\sin\theta$

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Exponential form

Multiplying Euler's formula by $r$ gives a third way to write a complex number, and the most compact one.

The three forms are three ways of writing the same number, and each suits a different task.

Euler's identity

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

Converting with technology

The GDC converts between forms directly, provided it is in radian mode and complex mode.

• On a TI-84 Plus CE: MODE, choose RADIAN and re^(θi) to show every complex result in exponential form (or a+bi for Cartesian). $\mathrm{e}^{\square}$ is 2nd LN, and $\mathrm{i}$ is 2nd .. Inside MATH → CMPLX, ▶Polar and ▶Rect convert a single result.

• On a Casio fx-CG50: in Run-Matrix, SET UP, set Complex Mode to r∠θ for polar results or a+bi for Cartesian, and Angle to Rad. OPTN → COMPLEX holds ►r∠θ and ►a+bi. The display $5.39\angle 2.76$ means modulus 5.39 and argument 2.76.

Products and quotients in polar form

Multiplying: multiply the moduli, add the arguments

The laws of exponents give the product rule at once.

Two results from subtopic 1.12 follow: $|zw| = |z||w|$, and $\arg(zw) = \arg z + \arg w$

Dividing: divide the moduli, subtract the arguments

Division undoes multiplication, so the moduli are divided and the arguments subtracted.

A special case is the reciprocal $\frac{1}{r\mathrm{e}^{\mathrm{i}\theta}} = \frac{1}{r}\mathrm{e}^{-\mathrm{i}\theta}$

Powers in polar form

De Moivre's theorem

Repeating the product rule $n$ times multiplies the modulus by itself $n$ times and adds the argument to itself $n$ times. In exponential form this is the law $(x^m)^n = x^{mn}$

The result also holds for negative integers. For $n = -1$ it gives $\frac{1}{r}\mathrm{e}^{-\mathrm{i}\theta}$, the reciprocal found in the division rule, and for $n = 0$

Powers that are real

The argument of $z^n$ is $n\theta$, so $z^n$ is real when $n\theta$ is a multiple of $\pi$

On an Argand diagram the powers $w, w^2, w^3, \ldots$ of a number with modulus greater than 1 lie on a spiral moving outwards, each one turned by the same angle from the last. With modulus exactly 1 the powers stay on the unit circle, and with modulus less than 1 they spiral in towards 0.

The geometry of complex numbers

Addition and subtraction as vectors

On an Argand diagram a complex number $a + b\mathrm{i}$ can be drawn as the arrow from the origin to $(a, b)$, the vector $\begin{pmatrix} a \\ b \end{pmatrix}$

Subtraction works the same way: $z - w$ is the arrow that goes from the tip of $w$ to the tip of $z$, because $w + (z - w) = z$

Multiplication as a rotation and a stretch

Multiplying $z$ by $w = r\mathrm{e}^{\mathrm{i}\phi}$ multiplies the modulus of $z$ by $r$ and adds $\phi$

The arithmetic agrees with the picture: $(2 + \mathrm{i})(1 + \mathrm{i}) = 2 + 2\mathrm{i} + \mathrm{i} + \mathrm{i}^2 = 1 + 3\mathrm{i}$

Dividing by $r\mathrm{e}^{\mathrm{i}\phi}$ reverses both effects: it rotates through $-\phi$ and scales by $\frac{1}{r}$. Taking the conjugate is a reflection in the real axis, which is not a rotation at all. Matrix transformations in subtopic 3.9 describe the same rotations and stretches with

Adding sinusoids of the same frequency

Alternating current

A battery pushes current in one direction at a steady voltage. A power station's generator instead produces a voltage that rises and falls smoothly, reversing direction many times each second.

In much of the world the mains supply has a nominal voltage of 230 V at a frequency of 50 Hz (50 complete cycles a second); in North America the usual combination is 120 V at 60 Hz. Mains voltages are quoted as root-mean-square values; the peak of a 230 V supply is $230\sqrt{2} \approx 325$ V. The model $V = A\cos(\omega t + \phi)$

The constant $\phi$ is the phase angle of the wave. Because $\omega t + \phi = \omega\left(t + \frac{\phi}{\omega}\right)$

Waves as real parts of complex numbers

Euler's formula gives $\mathrm{e}^{\mathrm{i}(\omega t + \phi)} = \cos(\omega t + \phi) + \mathrm{i}\sin(\omega t + \phi)$

As $t$ increases, $A\mathrm{e}^{\mathrm{i}(\omega t + \phi)}$ moves round a circle of radius $A$ at a steady rate, and the wave is the horizontal coordinate of that moving point. The fixed complex number $A\mathrm{e}^{\mathrm{i}\phi}$

The key step: the real part of a sum

Taking the real part of a sum gives the same result as adding the real parts, because $\operatorname{Re}(a + b\mathrm{i} + c + d\mathrm{i}) = a + c$. This allows two waves to be combined inside one complex expression.

For two waves of the same angular frequency $\omega$, the common factor $\mathrm{e}^{\mathrm{i}\omega t}$ comes out of the sum, leaving a sum of two phasors:

where $R\mathrm{e}^{\mathrm{i}\Phi} = A_1\mathrm{e}^{\mathrm{i}\phi_1} + A_2\mathrm{e}^{\mathrm{i}\phi_2}$

The method has four steps:

1. Write each wave as the real part of a complex exponential.

2. Combine them using $\operatorname{Re}(z_1) + \operatorname{Re}(z_2) = \operatorname{Re}(z_1 + z_2)$

The graphs of the two waves and their sum show this directly. All three have the same period, $\pi$, and the sum peaks between the peaks of the two parts.

Using technology for the phasor sum

With decimal phase angles the phasor sum is best done on the GDC: enter $A_1\mathrm{e}^{\phi_1\mathrm{i}} + A_2\mathrm{e}^{\phi_2\mathrm{i}}$

Sine waves, equivalent forms and cancellation

A sine wave is the imaginary part of the same exponential, because $\operatorname{Im}\left(\mathrm{e}^{\mathrm{i}\theta}\right) = \sin\theta$. Sums of sine waves of the same frequency are therefore found with $\operatorname{Im}(z_1) + \operatorname{Im}(z_2) = \operatorname{Im}(z_1 + z_2)$

The form $A\cos(Bt + C)$ is not unique. Adding $2\pi$ to $C$ changes nothing, and because $\cos(-x) = \cos x$

When two waves of equal amplitude are exactly half a cycle out of step, with phase difference $\pi$, their phasors are $A$ and $A\mathrm{e}^{\mathrm{i}\pi} = -A$, which sum to zero, so the waves cancel completely. Noise-cancelling headphones use this idea, producing a sound wave in antiphase with the unwanted noise so that the two add to almost nothing. With phase difference 0, the phasors point the same way and the amplitudes add, the largest possible result.

Case study: three-phase mains electricity

Power stations generate electricity on three wires, or phases, at once. The three voltages have the same amplitude and frequency, but their peaks are separated by a third of a cycle, $\frac{2\pi}{3}$ radians. In countries with a 230 V, 50 Hz supply, each phase measured against the neutral wire has peak voltage $230\sqrt{2} \approx 325$

Source: Wikipedia, "Mains electricity" and "Three-phase electric power" (230 V, 50 Hz; phases one-third of a cycle apart), checked 2026.

Each phase has a phasor of modulus 325: $325\mathrm{e}^{0\mathrm{i}}$, $325\mathrm{e}^{-\frac{2\pi}{3}\mathrm{i}}$

The sum of the three phases. Adding the phasors, $325\left(1 + \mathrm{e}^{-\frac{2\pi}{3}\mathrm{i}} + \mathrm{e}^{\frac{2\pi}{3}\mathrm{i}}\right) = 325\left(1 - \frac{1}{2} - \frac{\sqrt{3}}{2}\mathrm{i} - \frac{1}{2} + \frac{\sqrt{3}}{2}\mathrm{i}\right) = 0$

The voltage between two phases. An appliance connected between wires 1 and 2, instead of between one wire and neutral, receives $V_1 - V_2$. Its phasor is

because $\left|\frac{3}{2} + \frac{\sqrt{3}}{2}\mathrm{i}\right| = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3}$

Interpretation. The voltage between two phases is a sinusoid of the same 50 Hz frequency, with amplitude $\sqrt{3}$ times larger than each phase and a phase angle of $\frac{\pi}{6}$

Exam focus

Polar and Euler form is examined on Papers 1 and 2 (HL), often following a 1.12 part in the same question. Typical command terms are express, write down, find, plot, show that, describe and hence. Recurring shapes are: converting a number to $r\mathrm{e}^{\mathrm{i}\theta}$ exactly; products, quotients and powers in a stated argument range; the least $n$ for which a power is real; plotting successive powers on an Argand diagram; rotating a point to build a regular shape; and adding two voltages or currents of the same frequency into $A\cos(Bt + C)$

Write down the form you are converting to and the modulus and argument separately, the expression entered into the GDC (for example "$6 + 8\mathrm{e}^{-\mathrm{i}} = 12.3\mathrm{e}^{-0.578\mathrm{i}}$"), and in wave questions the step $\operatorname{Re}(z_1) + \operatorname{Re}(z_2) = \operatorname{Re}(z_1 + z_2)$

In (b), a candidate who multiplies Cartesian forms correctly earns the method mark, but the answer must be given in the requested exponential form with the argument in range. In (e), solving $50t - 0.503 = 2\pi$ gives a later maximum, not the first one.

When a question says "exact", the angle must be a multiple of $\pi$ and the modulus a surd: $\sqrt{2}\,\mathrm{e}^{-\frac{\pi}{4}\mathrm{i}}$

Practice questions

1. Express $-\sqrt{3} - \mathrm{i}$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$

Summary

• Polar form $z = r(\cos\theta + \mathrm{i}\sin\theta) = r\,\mathrm{cis}\,\theta$ and exponential form $z = r\mathrm{e}^{\mathrm{i}\theta}$

Self-check

• To do:

  I can write a complex number in polar and exponential form, with a positive modulus and the principal argument.

• To do:

  I can convert from polar or exponential form to Cartesian form, exactly for special angles and with decimals otherwise.

• To do:

  I can recognise when an expression such as $-2\,\mathrm{cis}\,\theta$ or $r(\cos\theta - \mathrm{i}\sin\theta)$ is not in polar form, and correct it.