Complex Numbers - Theory & Concepts

Complex Numbers

$Learning Objectives$

Identify the real and imaginary parts of a complex number and simplify powers of $i$ .
Add, subtract, multiply, and divide complex numbers in rectangular form.
Plot a complex number on an Argand plane and find its conjugate, modulus, and argument.
Convert reliably between rectangular and polar forms using the correct quadrant and angular units.
Apply De Moivre's theorem to integer powers, rotations, and all indexed $n$ -th roots.

Complex numbers extend the real number line to a two-dimensional plane. They occur in oscillations, alternating-current analysis, and polynomial equations with non-real roots. Unless otherwise stated, angles in the formulas below are in radians.

$Imaginary Unit and the Four-Step Cycle$

The imaginary unit satisfies $i^2=-1$ . In electrical engineering the same unit is often denoted by $j$ to avoid confusing it with current. The powers repeat every four exponents: $i^0=1$ , $i^1=i$ , $i^2=-1$ , $i^3=-i$ , and $i^4=1$ . For any integer $m$ , reduce $m$ modulo $4$ , including when $m$ is negative.

$Imaginary Unit$

i^2=-1,\qquad i^4=1

$Rectangular Form and Operations$

Write $z=a+bi$ for real $a$ and $b$ . Then $\operatorname{Re}(z)=a$ , $\operatorname{Im}(z)=b$ (not $bi$ ), and the conjugate is $\overline z=a-bi$ .

Add/subtract: $(a+bi)\pm(c+di)=(a\pm c)+(b\pm d)i$ .
Multiply: $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$ .
Divide: Multiply numerator and denominator by the denominator's conjugate; $(a+bi)(a-bi)=a^2+b^2=|z|^2$ .

The quotient is defined only when the denominator is nonzero.

Use the vector laboratory to compare componentwise addition with complex multiplication. Multiplication can alter both length and direction; it is not generally ordinary vector multiplication.

Complex Vector Arithmetic Plane

Arithmetic Operation

Vector

z_1 = a + bi

Real part (a)2

Imaginary part (b)3i

Vector

z_2 = c + di

Real part (c)1

Imaginary part (d)-2i

Algebraic Result Evaluation

z_1 + z_2 = (2 + 1) + (3 + -2)i = 3 + 1i

$Argand Plane, Magnitude, and Argument$

The horizontal coordinate is the real part $a$ ; the vertical coordinate is the imaginary part $b$ . For nonzero $z$ , the modulus $r=|z|=\sqrt{a^2+b^2}$ is its distance from the origin. The argument $\theta$ is the directed angle from the positive real axis. Arguments differing by $2\pi k$ represent the same nonzero point, where $k$ is an integer. The point $z=0$ has modulus zero and no defined argument.

$Read the Construction$

This is an analytical illustration: $z=-1+\sqrt3\,i$ lies in quadrant II, so $r=2$ and $\theta=120^\circ=2\pi/3$ radians. The dotted legs locate $a$ and $b$ ; the slanted arrow represents $z$ .

Change the coordinates in the interactive complex plane. Track what happens as the point crosses an axis; a single-argument inverse tangent must not be used without correcting the quadrant.

Modulus and Polar Conversion

Use atan2 or an equivalent quadrant-aware method for nonzero z.

r=\sqrt{a^2+b^2},\quad \theta=\operatorname{atan2}(b,a),\quad a=r\cos\theta,\quad b=r\sin\theta

Variables

Symbol	Description	Unit
$r$	Nonnegative modulus; strictly positive for a defined argument	-
$a$	Real Cartesian coordinate	-
$b$	Imaginary Cartesian coordinate	-
$\theta$	Chosen argument in radians, including correct quadrant	-

$Polar and Exponential Forms$

For $z\ne0$ , $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$ where $r>0$ and any coterminal $\theta$ gives the same point. A principal argument is commonly chosen in $(-\pi,\pi]$ , but always state the chosen branch. If a calculator is in degree mode, convert its angle to radians before using $e^{i\theta}$ or a radian formula.

$Multiplication Is Scaling Plus Rotation$

If $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2}$ , then $z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$ . Dividing nonzero complex numbers divides magnitudes and subtracts angles. Multiplying by $i=e^{i\pi/2}$ rotates a vector counterclockwise by $90^\circ$ without changing length.

$Rotation Interpretation$

The solid vectors have equal modulus $\sqrt2$ . Their arguments are $\pi/4$ and $3\pi/4$ , a difference of $\pi/2$ . The curved dashed arc indicates the direction of increasing argument, not a third vector.

De Moivre's Theorem

For integer n and a complex number written in polar form.

\big[r(\cos\theta+i\sin\theta)\big]^n=r^n\big(\cos(n\theta)+i\sin(n\theta)\big)

Variables

Symbol	Description	Unit
$r$	Modulus of z; require r > 0 for negative powers	-
$\theta$	Argument in radians	-
$n$	Integer exponent; n may be negative when z is nonzero	-
$i$	Imaginary unit satisfying i squared equals minus one	-

$All Distinct n-th Roots$

For a nonzero complex number $z=re^{i\theta}$ and positive integer $n$ , the solutions to $w^n=z$ are equally spaced by $2\pi/n$ on a circle of radius $r^{1/n}$ . The $n$ distinct roots use exactly $k=0,1,\ldots,n-1$ ; using $k=n$ repeats the first root. For $z=0$ , the sole distinct root is $w=0$ (of multiplicity $n$ ).

Finding n-th Roots

Use one angular unit consistently; radians are shown here.

w_k=r^{1/n}\left[\cos\left(\frac{\theta+2\pi k}{n}\right)+i\sin\left(\frac{\theta+2\pi k}{n}\right)\right],\quad k=0,1,\ldots,n-1

Variables

Symbol	Description	Unit
$w_k$	Indexed n-th root of nonzero z	-
$r$	Positive modulus of z	-
$n$	Positive integer root degree	-
$k$	Index from zero through n minus one	-
$\theta$	Any argument of z, in radians; branch choice permutes the roots	-

$Indexed Root Check$

The figure is a theoretical unit-circle construction for $w^4=1$ : the angular step is $2\pi/4=\pi/2$ ; raising each marked root to the fourth power returns $1$ . The vector lengths all equal $1$ .

Use the De Moivre explorer to vary the starting number and exponent. Compare the scaled radius $r^n$ with the rotated angle $n\theta$ , and check answers algebraically.

De Moivre's Theorem Explorer

Complex Number z

z = 1.20 \left(\cos 45^\circ + i \sin 45^\circ\right)

z = 1.20 e^{i 0.785}

Magnitude r (Radius)1.20

Angle θ (Degrees)45°

Power nn = 3

Theorem Application

Result:

z^{n}

z^{3} = 1.20^{3} \left(\cos (3 \cdot 45^\circ) + i \sin (3 \cdot 45^\circ)\right)

z^{3} = 1.728 \left(\cos 135^\circ + i \sin 135^\circ\right)

Notice that the magnitude raises geometrically to 1.728, while the angle multiplies linearly to 135° (or 135° coterminal).

z (Original)

z^{n}

(Result)

Intermediates

Key Takeaways

Powers of $i$ cycle every four exponents.
Rectangular operations combine real and imaginary components; division requires a nonzero denominator.
$|z|$ is a distance, while the argument is an angle undefined at the origin; always resolve its quadrant.
Products multiply moduli and add arguments; multiplication by $i$ rotates counterclockwise by $\pi/2$ .
De Moivre raises the modulus to $n$ and multiplies the argument by $n$ .
A nonzero number has $n$ distinct $n$ -th roots indexed $0$ through $n-1$ , equally spaced in argument.

$References$

OpenStax, Algebra and Trigonometry 2e, §10.5: Polar Form of Complex Numbers. Mathematical definitions, De Moivre's theorem, and root indexing; accessed October 2026.

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