The Complex Plane

The Argand Diagram #

A complex number needs a plane, not a line.

A complex number needs a plane rather than a line, because it has two parts

A real number needs one axis: each number is one position along a line.

A complex number has two parts, so it needs two directions: across and up.

Across is the real part, up is the imaginary part. That plane is the Argand diagram.

The Modulus of a Complex Number #

The distance from the origin to the point.

The modulus is the distance from the origin, so Pythagoras gives it

Draw the arrow from the origin. Its length is what the modulus measures.

The arrow steps across 3 and up 4, forming a right-angled triangle, so Pythagoras applies.

The modulus is $|z|=\sqrt{a^2+b^2}$|z| = √(a² + b²), straight from Pythagoras.

The Argument of a Complex Number #

The angle the arrow makes with the real axis.

The argument is the angle the arrow makes with the positive real axis

The arrow points in some direction. The angle it makes with the positive real axis is the argument.

Opposite over adjacent is imaginary over real, so $\tan \theta =\frac{b}{a}$tan θ = b/a.

The real and imaginary parts are equal, so the angle is 45° — the arrow lies along the diagonal.

Here tan still says 45°, but the arrow points into the second quadrant, so the angle is 180° − 45° = 135°.

Polar Form #

A modulus and an argument fix a complex number.

A modulus and an argument fix a complex number just as well as two parts do

Instead of a real part and an imaginary part, give the arrow a length and a direction: its modulus and its argument.

The real part is $r\cos \theta$r cos θ and the imaginary part is $r\sin \theta$r sin θ, so both forms agree.

Take (1 + i)(1 + i), already worked out as 2i: $\sqrt{2}\times \sqrt{2}=2$√2 × √2 = 2 and 45° + 45° = 90°.

The lengths multiply and the angles add every time, and that is what makes polar form useful.

The Form r(cos θ + i sin θ) #

The shorthand r at $\theta$θ, written properly.

Written out in full, a polar number is r times $\cos \theta$cos θ plus $i\sin \theta$i sin θ

In the Argand triangle, the horizontal leg is $r\cos \theta$r cos θ and the vertical leg is $r\sin \theta$r sin θ.

Factor out the r and the notation is $z=r\left(\cos \theta +i\sin \theta \right)$z = r(cos θ + i sin θ).

Books abbreviate it cis $\theta$θ — the c, i, s spell out cos, i, sin.

Dividing in Polar Form #

Divide the lengths, subtract the angles.

Dividing complex numbers divides the lengths and subtracts the angles

Multiplying multiplied the lengths and added the angles — dividing undoes both.

6 ÷ 2 = 3 and 90° − 30° = 60° — the quotient is r = 3 at 60°.

Check by multiplying back: 3 × 2 = 6 and 60° + 30° = 90°, the number we started with.

De Moivre’s Theorem #

A power multiplies the argument.

Raising to the power n multiplies the argument by n and raises the modulus to the power n

If multiplying adds the angles, then squaring must double the angle.

Multiply by z n times and both the length and the angle follow one rule.

De Moivre in one line: r at $\theta$θ, raised to n, is $r^n$rⁿ at $n\theta$nθ.

The Roots of Unity #

Three answers to $z^3=1$z³ = 1, spaced round a circle.

$z^n=1$zⁿ = 1 has n answers, spaced evenly round the unit circle

$z^3=1$z³ = 1 has three answers, spaced evenly round the unit circle: 1, $\omega$ω and ${\omega }^2$ω².

Cube the 120° arrow: the angle triples to 360° — a full turn, landing on 1.

Exactly three angles triple to whole turns, so $z^3=1$z³ = 1 has exactly three roots.

The five fifth roots of 1 sit 360° ÷ 5 = 72° apart. The n-th roots of 1 split the circle into n equal steps.

Regular Polygons from the Roots of Unity #

The vertices are the roots, so the answers follow.

The n-th roots of unity are a regular n-gon, so its coordinates, its sum and its side come straight off them

The six roots of $z^6=1$z⁶ = 1 are the vertices of a regular hexagon on the unit circle.

Each vertex is the cosine and sine of its own angle, so no measuring is needed.

The roots add to zero: sum the series and the numerator is ${\omega }^6-1$ω⁶ − 1, which is 0.

A side joins two neighbors across a central angle of $\frac{360^{\circ }}{n}$360°/n, so its length is $|\omega -1|=2\sin \left(\frac{180^{\circ }}{n}\right)$|ω − 1| = 2 sin(180°/n).

Scale the circle and the same formula still works: $z^4=16$z⁴ = 16 gives a square of side $2\sqrt{2}$2√2.

Multiple Angles by De Moivre #

Expand the power, then match the parts.

Expanding De Moivre and matching the real parts writes $\cos n\theta$cos nθ in powers of $\cos \theta$cos θ

De Moivre says the cube of $\cos \theta +i\sin \theta$cos θ + i sin θ is $\cos 3\theta +i\sin 3\theta$cos 3θ + i sin 3θ. Now expand it.

Expand by the binomial theorem and sort the terms into a real part and an i part.

Two complex numbers are equal only if both parts match, so one line gives two results.

Replace $s^2$s² with $1-c^2$1 − c² and only cosines are left: $\cos 3\theta =4{\cos }^3\theta -3\cos \theta$cos 3θ = 4cos³θ − 3cos θ.

The same move at n = 2 produces the double-angle formulas.

The Identity z + 1/z = 2 cos θ #

A number added to its own reciprocal.

For a unit-length z the sum $z+\frac{1}{z}$z + 1/z is twice the cosine of its argument

Take z of length 1. De Moivre at n = −1 turns the angle back, giving the conjugate.

So $\frac{1}{z}$1/z is the mirror image of z in the real axis, sitting the same distance out.

Add them and the i parts cancel; subtract them and the real parts cancel. That gives two identities.

Every power works the same way: $z^n+\frac{1}{z^n}$zⁿ + 1/zⁿ is $2\cos n\theta$2 cos nθ, for any whole n.

Cube $z+\frac{1}{z}$z + 1/z, pair the outer terms and the inner ones, and a power of $\cos \theta$cos θ turns into multiple angles.

Summing a Series with De Moivre #

A cosine sum with a geometric one inside.

A cosine series is the real part of a geometric series in one complex number

Neither of these sums is geometric on its own: cosines have no common ratio.

Form C + iS, and every term becomes a power of one number z.

That is geometric with ratio z, so the ordinary sum formula finishes it in one line.

Split the quotient back into parts: the real part is C and the i part is S.

Check it at $\theta =90^{\circ }$θ = 90°: the powers of i sum to 0, and so do the cosines.

The Exponential Form of a Complex Number #

The exponential series, fed an imaginary number.

The series for e to the $i\theta$iθ is exactly $\cos \theta$cos θ plus $i\sin \theta$i sin θ

The exponential series works for any input. Substitute $x=i\theta$x = iθ and see what the powers become.

The powers of i cycle, so the terms sort into two alternating series.

Those two series are the Maclaurin series for cosine and for sine. That is the identity.

So $e^{i\theta }$e^(iθ) always has length 1: it is the point at angle $\theta$θ on the unit circle.

Multiply by r and every complex number is written re to the $i\theta$iθ.

Now lengths multiply and angles add because that is what index laws already do.

The nth Roots of a Complex Number #

One number, n roots, evenly spaced.

Every non-zero complex number has n distinct nth roots, evenly spaced round a circle

One point on the plane has many angles: adding a whole turn changes nothing.

Divide each of those angles by n and they stop agreeing, so n different roots appear.

Take the cube roots of 8i: each has length 2, and the angles are 30°, 150° and 270°.

The three roots sit on one circle of radius 2, a third of a turn apart.

Taking k = n adds a whole turn and repeats a root, which is why there are only n of them.

Circles and Bisectors on the Argand Diagram #

A modulus is a distance, so it draws a shape.

A modulus is a distance, so an equation in moduli names a set of points on the plane

|z| is the distance from z to the origin. Fixing it at 2 draws a circle of radius 2.

|z − a| is the distance from z to a, so the same circle moves to be centered at a.

Read it straight off: whatever z is measured from is the center, and r is the radius.

|z − a| = |z − b| asks for equal distances: the perpendicular bisector of the segment ab.

Fix one distance and you get a circle; set two distances equal and you get a straight line.

Rays and Regions on the Argand Diagram #

An argument picks a direction, not a place.

Fixing the argument draws a half-line, and turning an equation into an inequality shades a region

arg(z − a) is the direction from a to z. Fixing it at 45° gives a half-line from a.

Only half of the line qualifies: the opposite direction from a points at −135° instead.

Loosen |z| = 3 to $|z|\le 3$|z| ≤ 3 and the locus grows from the rim to the whole disc.

A strict inequality leaves the rim out, so the boundary is drawn dashed to show it is excluded.

Nearer to 2i than to −2i means one side of the perpendicular bisector: everything above the real axis.