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A complex number as a distance and a turn: $z = re^{i\theta}$, multiplication as adding arguments and multiplying moduli, de Moivre's theorem for powers, and the quadrant correction that converting back demands.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to convert between rectangular and polar form in both directions with the quadrant correction, state what multiplication, division, conjugation and powers do to a modulus and to an argument, apply de Moivre's theorem to compute a high power in one line, and say why the principal argument has to be brought back into range after arguments are added.
The modulus as a distance from the origin, and the observation that multiplying by the imaginary unit is a quarter turn. Polar form is what happens when that observation is taken seriously: if multiplication turns points, then a complex number should be described by how far out it is and how far round.
An argument of a non-zero $z$ is any angle $\theta$ with $z = |z|(\cos\theta + i\sin\theta)$; there are infinitely many, differing by whole turns. The principal argument $\operatorname{Arg} z$ is the one in $(-\pi, \pi]$ — greater than minus half a turn and at most half a turn. Polar form is $z = re^{i\theta}$ with $r = |z|$. De Moivre's theorem is the statement $\left(e^{i\theta}\right)^{n} = e^{in\theta}$. The number $0$ has a modulus and no argument at all.
Every non-zero complex number can be written
$$z = r(\cos\theta + i\sin\theta) = re^{i\theta}, \qquad r = |z| > 0,$$
and the pair $(r, \theta)$ describes it completely — except that $\theta$ is fixed only up to adding whole turns, which is the one complication of the whole subject and never goes away.
Why this form is worth the conversion. Multiply two numbers in polar form:
$$r e^{i\theta} \cdot s e^{i\varphi} = rs\, e^{i(\theta + \varphi)}.$$
The moduli multiply and the arguments add, and the two do not interfere. Division divides and subtracts; conjugation keeps the modulus and negates the argument; and repeating multiplication $n$ times gives de Moivre's theorem
$$z^{n} = r^{n}e^{in\theta}.$$
In rectangular form, computing $(1 + i)^{10}$ means ten multiplications or a binomial expansion; in polar form it is $2^{5}e^{i\,10\pi/4}$ and takes a line.
Converting. From polar to rectangular is a substitution. From rectangular to polar, $r = \sqrt{x^{2} + y^{2}}$ is immediate, but the angle needs care: $\tan\theta = y/x$ does not distinguish a point from the point directly opposite it, so the inverse tangent must be corrected for the quadrant. A point in the second or third quadrant needs half a turn added to what the calculator returns.
And the identity everything rests on. $e^{i\theta} = \cos\theta + i\sin\theta$ is a definition for now and a theorem in two lessons' time, when the exponential is defined by its series. Setting $\theta = \pi$ gives $e^{i\pi} + 1 = 0$.
Another way: picture
Think of a complex number as an instruction: turn through $\theta$, then walk out a distance $r$. Multiplying two numbers composes the two instructions — turn through both angles, walk out the product of the distances. Squaring doubles the angle and squares the distance, so a point on the unit circle stays on it while sliding twice as far round, and a point outside runs away much faster than it turns.
Another way: steps
The questions in this lesson ask for angles in degrees, and the rest of the course works in radians. That is a deliberate compromise and it is worth saying why, because it is a place where the exercises and the mathematics genuinely differ.
In radians, almost every argument worth asking about is an irrational multiple of $\pi$: the argument of $1 + i$ is $\pi/4$, of $-1$ is $\pi$, of $1 + i\sqrt{3}$ is $\pi/3$. None of those can be typed into a box as a terminating number, and a question whose answer cannot be written down is not a question. In degrees the same three answers are $45$, $180$ and $60$ — whole numbers, exactly right, and typed in a second.
So: degrees when an angle is the answer, radians when an angle is inside a formula. Where an answer genuinely is a multiple of $\pi$ — and later in this course a great many of them are — the question will ask for the multiplier instead, which is the same dodge wearing different clothes. An argument is fixed only up to whole turns, and every strangeness of the logarithm, the power and the root comes from that one sentence. Choosing a branch is choosing which turn to call the right one, and the cut is where the choice comes apart. A calculation that treats an argument as a single number is a calculation that will contradict itself the first time it crosses the cut.
Almost every strange thing in this course traces back to one sentence: the argument is defined only up to adding whole turns. Three consequences catch people out.
Arguments add, but the sum may leave the principal range. Two numbers at $100^\circ$ multiply to give one at $200^\circ$, which as a principal argument is $-160^\circ$. The arithmetic is right; the answer needs bringing home.
$\operatorname{Arg}(zw)$ is not always $\operatorname{Arg} z + \operatorname{Arg} w$. The example above is a counterexample. It is the argument, not the principal argument, that adds, and the difference is exactly a whole turn.
Zero has no argument. Every angle would do, so none is picked out, and any formula involving $\operatorname{Arg} 0$ is meaningless rather than merely awkward. This is why the logarithm will have nothing to say at the origin.
$(1 + i)^{10}$: the modulus of $1 + i$ is $\sqrt{2}$ and its argument is $45^\circ$.
Convert first.
De Moivre gives modulus $(\sqrt{2})^{10} = 32$ and argument $10 \times 45 = 450^\circ$, which is $90^\circ$ after removing a whole turn.
Moduli multiply, arguments add.
Modulus $32$ at $90^\circ$ is $32i$.
Convert back at the end.
Find the principal argument of $-3 - 3i$. The ratio of the parts is $(-3)/(-3) = 1$, and the inverse tangent of $1$ is $45^\circ$.
The ratio has lost both minus signs.
But the point is in the third quadrant, down and to the left, so the true angle is $45^\circ - 180^\circ = -135^\circ$.
Correct for the quadrant, then bring into the principal range.
The point is two units straight down, so the modulus is $2$.
Read it off the picture.
Straight down is a quarter turn clockwise, so the principal argument is $-90^\circ$.
$z$ has modulus $6$ and argument $26^\circ$; $w$ has modulus $3$ and argument $17^\circ$. Fill in the table, giving every angle in degrees.
| Value | |
|---|---|
| The modulus of the product | |
| The argument of the product | |
| The modulus of the square of the first number | |
| The argument of the square of the first number | |
| The modulus of the conjugate of the first number | |
| The argument of the conjugate of the first number |
$z$ has modulus $7$ and argument $32^\circ$. What is the argument of $z^{2}$, in degrees?
Answer:
Match each number to its principal argument in degrees.
| $0^\circ$ | $90^\circ$ | $180^\circ$ | $-90^\circ$ | |
|---|---|---|---|---|
| $8$ | ||||
| $8i$ | ||||
| $-8$ | ||||
| $-8i$ |
Put the steps of writing $4 + 9i$ in polar form in order.
Number the steps in order (write the number in the box):
$z$ has modulus $2$ and argument $64^\circ$. What is the modulus of $z^{3}$?
Answer:
$z$ has modulus $1$ and argument $210^\circ$. What is the smallest positive integer $n$ with $z^{n} = 1$?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
$z$ has modulus $4$ and argument $60^\circ$; $w$ has modulus $5$ and argument $26^\circ$. Fill in the table, giving every angle in degrees.
| Value | |
|---|---|
| The modulus of the product | |
| The argument of the product | |
| The modulus of the square of the first number | |
| The argument of the square of the first number | |
| The modulus of the conjugate of the first number | |
| The argument of the conjugate of the first number |
You can move between rectangular and polar form, and you can say what a product does to each of the two quantities separately. Say in your own words why the argument of a number is not a single number. Next: the roots of unity, which are what happens when de Moivre's theorem is run backwards.
9. Your turn: modulus and principal argument of $-2i$, step 2