Complex Numbers

Concept

Argand Diagram

Every complex number $z = a + bi$ corresponds to a unique point $(a, b)$ in the plane: $a$ along the horizontal real axis, $b$ along the vertical imaginary axis.

Re axisHorizontal axis. Carries the real part $a$.

Im axisVertical axis. Carries the imaginary part $b$.

Modulus$|z| = \sqrt{a^2+b^2}$, the distance from the origin to the point.

Argument$\arg(z) = \theta$, the angle the arrow makes with the positive real axis.

$z = 3 + 2i$

Interactive

Click anywhere to place z

Argand diagram · click or drag the point

        |z| = 3.61   arg(z) = 0.59 rad
      

Concept

Arithmetic with i² = −1

The binomial form $a+bi$ is the natural form for addition and subtraction, and it can still handle multiplication and division: treat $i$ like a variable, then replace $i^2$ with $-1$ wherever it appears.

+

$(a+bi) + (c+di) = (a+c) + (b+d)i$

−

$(a+bi) - (c+di) = (a-c) + (b-d)i$

×

$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

÷

$\dfrac{a+bi}{c+di} = \dfrac{(a+bi)(c-di)}{c^2+d^2}$, multiply by the conjugate of the denominator

Interactive

Combine two complex numbers

z1 blue · z2 pink · result gold

z1 = a + bi

a =2.0

b =1.0

z2 = c + di

c =-1.0

d =2.0

(2 + 1i) + (-1 + 2i) = 1 + 3i

Concept

z = r(cosθ + i sinθ) = re^iθ

Instead of describing $z$ by its horizontal and vertical parts, polar form describes it by its distance from the origin, $r = |z|$, and its direction, $\theta = \arctan\left(\dfrac{b}{a}\right)$, adjusted for the quadrant of $z$ (section 04).

To polar$r = \sqrt{a^2+b^2}$
$\theta = \arctan\left(\dfrac{b}{a}\right)$, adjusted for the quadrant

To binomial$a = r\cos\theta$
$b = r\sin\theta$

Euler's formula$e^{i\theta} = \cos\theta + i\sin\theta$
so $re^{i\theta}$ and $r(\cos\theta+i\sin\theta)$ are the same number: the exponential form is just a compact way to write the polar form.

Best forMultiplication, division, powers and roots (section 06).

Interactive

z = re^iθ on the unit circle

blue = r cosθ · pink = r sinθ

r =2.00

θ =3π/4

z = 2.00 e^{i · 3π/4} = -1.41 + 1.41i

Memory trick

Five Fingers, Five Angles

Five rays extend from the origin, from the horizontal axis up to the vertical, one for each landmark angle between 0 and $\pi/2$. Pick one of the five rays as your angle $\theta$, then:

sin

Count the rays below the one you picked: $\sin\theta = \dfrac{\sqrt{\text{below}}}{2}$

cos

Count the rays above the one you picked: $\cos\theta = \dfrac{\sqrt{\text{above}}}{2}$

tan

Combine both counts: $\tan\theta = \dfrac{\sqrt{\text{below}}}{\sqrt{\text{above}}}$

Other quadrants

Quadrant II$\theta = \pi - \theta_{ref}$: sine keeps its sign, cosine flips.

Quadrant III$\theta = \pi + \theta_{ref}$: both sine and cosine flip sign.

Quadrant IV$\theta = 2\pi - \theta_{ref}$: cosine keeps its sign, sine flips.

Interactive

Pick an angle, read the values

unit circle

sin(π/3) = √3/2 (0.87)
cos(π/3) = 1/2 (0.50)
tan(π/3) = √3 (1.73)

Example

Angle → sin, cos, tan

1

Take $\theta = \dfrac{5\pi}{6}$. It is past $\pi/2$, so find its reference angle inside quadrant I: $\pi - \dfrac{5\pi}{6} = \dfrac{\pi}{6}$.

2

$\dfrac{5\pi}{6}$ sits between $\pi/2$ and $\pi$: quadrant II. There, sine is positive and cosine is negative.

3

$\sin\dfrac{5\pi}{6} = \sin\dfrac{\pi}{6} = \dfrac{1}{2}$
$\cos\dfrac{5\pi}{6} = -\cos\dfrac{\pi}{6} = -\dfrac{\sqrt3}{2}$

4

$\tan\dfrac{5\pi}{6} = \dfrac{\sin(5\pi/6)}{\cos(5\pi/6)} = -\dfrac{1}{\sqrt3} = -\dfrac{\sqrt3}{3}$

Example

tan → angle, with quadrant

This is the step most often needed to turn a binomial form into an exponential one.

1

Start from $z = -1 + \sqrt3\,i$, so $a = -1$, $b = \sqrt3$.

2

$\tan\theta = \dfrac{b}{a} = \dfrac{\sqrt3}{-1} = -\sqrt3$
Ignoring the sign for a moment, $\arctan(\sqrt3) = \pi/3$ is the reference angle.

3

$a < 0$ and $b > 0$ place $z$ in quadrant II, so $\theta = \pi - \pi/3 = 2\pi/3$.

4

$r = |z| = \sqrt{a^2+b^2} = \sqrt{1+3} = 2$
so $z = 2\,e^{i \cdot 2\pi/3}$.

Concept

Conjugate

The conjugate of $z = a+bi$ flips the sign of the imaginary part: $\bar z = a - bi$. Geometrically, it is the reflection of $z$ across the real axis.

Modulus$z \bar z = a^2+b^2 = |z|^2$, always a non negative real number.

Real part$z + \bar z = 2\,\text{Re}(z)$

Imag part$z - \bar z = 2i\,\text{Im}(z)$

Sums$\overline{z_1+z_2} = \bar z_1 + \bar z_2$

Products$\overline{z_1 z_2} = \bar z_1 \bar z_2$

DivisionDividing by $z$ means multiplying by $\bar z / |z|^2$, which is how section 02's division rule arises.

Interactive

z and its mirror image

z red · conj(z) green

a =2.0

b =1.5

Concept

zⁿ and the n roots

Powers and roots are effortless in exponential form, since raising to a power just scales the angle.

Power

$z^n = r^n e^{in\theta}$
$= r^n(\cos n\theta + i \sin n\theta)$

Roots

If $z^n = r\,e^{i\theta}$, the $n$ solutions satisfy:
modulus: $|z| = \sqrt[n]{r}$
angle: $\theta_k = \dfrac{\theta + 2\pi k}{n}$, for $k = 0, 1, \dots, n-1$

Shape

The $n$ roots sit at equal angular spacing of $2\pi/n$ radians, all at the same distance $r^{1/n}$ from the origin: the corners of a regular polygon.

Interactive

n-th roots of z = r e^iθ

roots on a circle of radius r^(1/n)

n =5

r =1.0

θ =0

5 roots, spaced 1.26 rad apart

Example

Raising to a power

1

Start from $z = 1+i$. In exponential form, $r=\sqrt2$, $\theta=\pi/4$, so $z = \sqrt2\,e^{i\pi/4}$.

2

Apply De Moivre for $z^6$:
$z^6 = (\sqrt2)^6\,e^{i\cdot 6\pi/4} = 8\,e^{i\cdot 3\pi/2}$

3

Convert back: $\cos(3\pi/2)=0$, $\sin(3\pi/2)=-1$
so $z^6 = 8(0 - i) = -8i$.

Example

Solving z⁵ = 32e^i·π

1

Write the right-hand side in exponential form: $w = 32\,e^{i\pi}$, so $r=32$, $\theta=\pi$.

2

The radius of every root is $r^{1/5} = 32^{1/5} = 2$.

3

The angles are $\theta_k = \dfrac{\pi + 2\pi k}{5}$ for $k=0,1,2,3,4$:
$\dfrac{\pi}{5}, \dfrac{3\pi}{5}, \pi, \dfrac{7\pi}{5}, \dfrac{9\pi}{5}$

4

So the five solutions are $z_k = 2\,e^{i\theta_k}$: five points at radius 2, spaced $2\pi/5$ rad apart, as in the interactive above (try $n=5$, $r=2$, $\theta=\pi$).

	Binomial	Exponential
Written as	$a + bi$	$re^{i\theta}$
Parameters	real part, imaginary part	modulus, argument
Best for	addition, subtraction	multiplication, powers, roots
Multiplying	expand and use $i^2=-1$	multiply moduli, add exponents
Example	$1 + \sqrt3\,i$	$2e^{i\pi/3}$

The Complex Plane

Binomial Form and Operations

Polar and Exponential Form

Sine and Cosine of Common Angles

Conjugate and Properties

Powers and Roots (De Moivre)

Summary