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

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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.

Moving Points on the Argand Diagram #

Turn, flip or slide z with one operation.

Multiplying by i turns z a quarter turn about O, negating it a half turn, and conjugating reflects it in the real axis

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$i\left(3+i\right)=3i+i^2=-1+3i$i(3 + i) = 3i + i² = −1 + 3i: multiplying by i turns z a quarter turn counterclockwise about O.

−z = −3 − i is a half turn about O: both parts change sign, and the arrow points the opposite way.

The conjugate z* = 3 − i changes only the sign of the imaginary part: a reflection in the real axis.

Adding w = 1 + 2i moves z by the same step as the arrow of w: 1 across and 2 up.

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

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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

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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

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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

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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

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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

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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

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$z^3=1$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

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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

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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

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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

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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

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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

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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

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|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

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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.