🌀The Complex Plane & Polar Form

Herald Vex widened the number line into a + bi. Now he widens it once more, and this time the picture does the work.

A complex number needs two real numbers to pin it down, so one line is not enough — but a plane is. Draw the ordinary x-axis and call it the real axis, draw the y-axis and call it the imaginary axis, and then

a + bi is the point (a, b)

That is the whole construction. 3 - 4i is the point (3, -4), one step right of centre and four steps down. Real numbers still live where they always did, along the horizontal axis, because 6 is the point (6, 0). Pure imaginary numbers stand on the vertical axis: 4i is (0, 4).

The plane is called the complex plane, or the Argand diagram, and its quadrants are numbered the usual way — first quadrant right and up, then anticlockwise.

It is often more useful to draw a complex number as an arrow from the origin out to its point rather than as a dot. An arrow has two things a dot does not: a length and a direction. Those two numbers are the whole of the rest of this lesson.

Drawn this way, three pieces of algebra you already know turn into geometry:

• ◆the conjugate a - bi is z reflected in the real axis;
• ◆-z is z turned through half a turn about the origin;
• ◆z + w is the arrow for w laid tip-to-tail on the end of the arrow for z.

That last one is why adding complex numbers adds the parts separately: you go along, then up, then along again, then up again, and the order does not matter.

The modulus of z = a + bi, written |z|, is the length of that arrow. The arrow, the real axis and a vertical line make a right-angled triangle with legs |a| and |b|, so Pythagoras settles it:

|z| = sqrt(a^2 + b²)

So |3 - 4i| = sqrt(9 + 16) = sqrt(25) = 5. The signs vanish the instant you square, which is exactly right: a length is never negative.

Because the modulus is a length, the distance between two points on the complex plane is

|z - w|

Subtract first, take the modulus second. |z| - |w| is a different number and usually the wrong one.

The argument of z, often written theta, is the direction of the arrow: the angle turned anticlockwise from the positive real axis. It is reported on the interval from 0° up to (but not including) 360°.

Finding it takes two steps, and skipping the second is the classic slip. First, the reference angle — the acute angle the arrow makes with the real axis — comes from

tan(reference) = |b| / |a|

Second, the quadrant places it. A calculator's arctan only ever answers between -90° and 90°, so it can only be trusted in the first and fourth quadrants:

• ◆first quadrant: theta = reference
• ◆second quadrant: theta = 180° - reference
• ◆third quadrant: theta = 180° + reference
• ◆fourth quadrant: theta = 360° - reference

Look at the point before you look at the calculator. -1 + i is up and to the left, so its argument has to be somewhere between 90° and 180°; the reference angle is 45°, so theta = 135°. A calculator handed -1/1 would have said -45°, which is a real direction — just the direction of the opposite arrow, 1 - i.

A few directions are worth knowing without any working: along the positive real axis is 0°, straight up is 90°, along the negative real axis is 180°, straight down is 270°.

If you know how far out (r) and in which direction (theta), you can rebuild the number, because the triangle gives

a = r cos theta and b = r sin theta

Substituting those into a + bi gives the polar form, also called the trigonometric form:

z = r(cos theta + i sin theta)

Note the plus sign in the middle. It is always a plus; a minus there is describing the conjugate.

Rectangular to polar. Find r by Pythagoras, find theta by the reference angle plus the quadrant. So for z = -3 + 3i: r = sqrt(9 + 9) = sqrt(18) = 3 sqrt(2), the reference angle is 45°, and the point is up and to the left, so theta = 135°. Then z = 3 sqrt(2)(cos 135° + i sin 135°).

Polar to rectangular. Multiply out, using the unit-circle values you already know:

cos 30° = sqrt(3)/2 sin 30° = 1/2 cos 45° = sqrt(2)/2 sin 45° = sqrt(2)/2 cos 60° = 1/2 sin 60° = sqrt(3)/2

So 4(cos 60° + i sin 60°) has real part 4 * 1/2 = 2 and imaginary part 4 * sqrt(3)/2 = 2 sqrt(3), which makes it 2 + 2 sqrt(3)i. Leave the surd exactly as it is: rounding here throws away the very thing the exact form was for.

The same conversion turns rectangular coordinates (x, y) into polar coordinates (r, theta) and back. The complex plane and the coordinate plane are the same sheet of paper with different labels.

Adding is easy in rectangular form and horrible in polar form. Multiplying is the other way round, and multiplying is what this topic is really about.

Multiply two numbers in polar form, expand, and the sum and difference formulas for cosine and sine collapse the whole thing into one line:

r(cos A + i sin A) * s(cos B + i sin B) = rs(cos(A + B) + i sin(A + B))

In words: multiply the moduli, add the arguments. Multiplying by a complex number stretches the plane by r and rotates it by theta.

Dividing undoes that, so it divides the moduli and subtracts the arguments:

z / w = (r/s)(cos(A - B) + i sin(A - B))

If the new argument falls outside 0° to 360°, add or take away a full turn until it is back inside. 40° - 100° = -60°, which is the same direction as 300°.

Multiply a number by itself n times and the rule applies n times. That is De Moivre's theorem:

(r(cos theta + i sin theta))^n = rⁿ(cos(n theta) + i sin(n theta))

The modulus is raised to the power and the argument is multiplied by it — both, not one. Try (1 + i)⁸ by expanding brackets and you will be there all evening; in polar form, 1 + i is sqrt(2)(cos 45° + i sin 45°), so the eighth power has modulus (sqrt(2))^8 = 16 and argument 8 * 45° = 360°, which is the direction 0°. The answer is 16, and it took one line.

On the real number line, 8 has one cube root. On the complex plane it has three, and every non-zero complex number has exactly n nth roots.

Run De Moivre backwards. If wⁿ = z, then |w|ⁿ = |z| and n * arg(w) = arg(z) plus any whole number of full turns. So:

modulus of every root = the real nth root of |z| arguments = (arg(z) + 360°k) / n, for k = 0, 1, 2, ..., n - 1

Every root has the same modulus, so they all sit on one circle centred at the origin. Their arguments differ by 360°/n, so they are spread evenly round it — which means the n roots are the vertices of a regular polygon: an equilateral triangle for cube roots, a square for fourth roots, a regular pentagon for fifth roots.

Take the fourth roots of -16. In polar form -16 is 16(cos 180° + i sin 180°). The modulus of each root is the real fourth root of 16, which is 2, and the arguments are (180° + 360°k)/4 = 45°, 135°, 225°, 315°. Four roots, 90° apart, at distance 2 from the origin — a square. Once you have the first one, you never work out another from scratch; you just keep adding 90°.