Part V · Complex Numbers, Trigonometry, and Advanced Functions

Chapter 14 — Complex Numbers and Polar Representation

Why This Matters

The equation $$x^2+1=0$$ has no real solution, yet such equations arise naturally when we analyse oscillation. Complex numbers enlarge the number system in a controlled way. Their geometry also provides the cleanest language for amplitude and phase in sinusoidal steady-state analysis.

Complex numbers turn AC circuit trigonometry into algebra. In AC circuits, voltages and currents oscillate at frequency $\omega$ . Phasors represent these sinusoidal signals as complex numbers $V = |V| e^{i\phi}$ , turning differential equations into simple Ohm's-law-like algebraic relations.

Algebraic rules of the imaginary unit

Starting point. Define $$i$$ by $$i^2 = -1$$ (or $$j^2 = -1$$ in electronics notation).

\[i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1\]
Powers of $$i$$ form a 4-cycle representing repeated $90^\circ$ rotations in the complex plane.reversible
\[(a + bi)(c + di) = (ac - bd) + (ad + bc)i\]
Multiply algebraically and use $$i^2 = -1$$ to group real and imaginary parts.reversible
\[\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}\]
Multiply numerator and denominator by the complex conjugate $$c - di$$ to make the denominator real.reversible

Conclusion. The complex plane extends the real numbers into a 2D field. Every non-zero complex number has an exact multiplicative inverse: $\frac{1}{z} = \frac{\bar{z}}{|z|^2}$ .

To add/subtract: $(a+bi) \pm (c+di) = (a\pm c) + (b\pm d)i$ (componentwise).
To multiply: expand with FOIL and substitute $$i^2 = -1$$ .
To divide: multiply top and bottom by the complex conjugate $\bar{z} = c - di$ .
Modulus: $|a+bi| = \sqrt{a^2 + b^2}$ . Conjugate: $\overline{a+bi} = a - bi$ .

14.1 The complex number system

Define $$i$$ by $$i^2=-1$$ . A complex number is

\[z=x+iy,\]

where $$x$$ and $$y$$ are real. The real and imaginary parts are $\operatorname{Re}z=x$ and $\operatorname{Im}z=y$ . Two complex numbers are equal only when both corresponding parts are equal.

Addition and multiplication follow algebra, with $$i^2=-1$$ :

\[(a+ib)(c+id)=(ac-bd)+i(ad+bc).\]

The conjugate of $$z=x+iy$$ is $\bar z=x-iy$ . Multiplying conjugates removes the imaginary cross terms:

z\bar z=x^2+y^2.

This motivates the modulus

|z|=\sqrt{z\bar z}=\sqrt{x^2+y^2}.

For $z\ne0$ ,

\frac1z=\frac{\bar z}{|z|^2}.

Example — division by a complex number

\frac{3+4i}{1-2i}=\frac{(3+4i)(1+2i)}{(1-2i)(1+2i)} =\frac{-5+10i}{5}=-1+2i.

The conjugate is chosen because it makes the denominator real and positive.

Complex arithmetic is pair arithmetic with one new rule. Write a complex number as an ordered pair of real and imaginary coordinates, then impose i²=-1. Addition combines coordinates; multiplication follows expansion and collects real and imaginary parts. The plane picture is a consequence of this algebra, not an unrelated diagram.

A complex number as a point and vector with real and imaginary coordinates.

14.2 Complex plane, arguments, and radians

Plot $$z=x+iy$$ as the point $$(x,y)$$ in a plane: horizontal coordinate is real part, vertical coordinate is imaginary part. Its distance from the origin is $$|z|$$ . An angle from the positive real axis to the point is an argument.

Arguments are not unique: if $\theta$ is an argument, so is $\theta+2\pi k$ for every integer $$k$$ . A chosen principal argument $\operatorname{Arg}z$ uses one stated interval, commonly $(-\pi,\pi]$ .

Radians are defined by arc length divided by radius. On a unit circle, an angle $\theta$ has arc length $\theta$ . This is why derivative and power-series formulas for sine and cosine use radians rather than degrees.

Cartesian versus polar representation. A complex number $$z = a + bi$$ specifies coordinates $$(a, b)$$ on the real and imaginary axes; polar form $z = r e^{i\theta}$ specifies distance $$r = |z|$$ from the origin and counterclockwise angle $\theta = \operatorname{Arg}(z)$ from the positive real axis.

Converting between Cartesian and polar forms

Starting point. Let $$z = x + iy$$ with $x, y \in \mathbb{R}$ .

\[r = |z| = \sqrt{x^2 + y^2}\]
The modulus $$r$$ is the Euclidean distance from $$(0, 0)$$ to $$(x, y)$$ .reversible
\[x = r \cos\theta, \quad y = r \sin\theta\]
Project polar vector $(r, \theta)$ onto the real and imaginary axes.reversible
\[z = r(\cos\theta + i\sin\theta) = r e^{i\theta}\]
Combine using Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ .reversible

Conclusion. Cartesian form $$x + iy$$ is ideal for addition and subtraction. Polar form $r e^{i\theta}$ is ideal for multiplication, division, and powers.

Compute modulus: $r = \sqrt{x^2 + y^2}$ .
Compute reference angle: $\alpha = \arctan(|y|/|x|)$ .
Determine quadrant and set $\theta = \operatorname{Arg}(z) \in (-\pi, \pi]$ :
Quadrant I: $\theta = \alpha$ ; Quadrant II: $\theta = \pi - \alpha$ ; Quadrant III: $\theta = -\pi + \alpha$ ; Quadrant IV: $\theta = -\alpha$ .

14.3 Polar coordinates: converting points and checking quadrants

The same geometry describes an ordinary Cartesian point $$(x,y)$$ and a complex number $$z=x+iy$$ . Its polar coordinates are a radius $r\ge0$ and an angle $\theta$ measured from the positive real axis:

x=r\cos\theta,\qquad y=r\sin\theta,\qquad r=\sqrt{x^2+y^2}.

For $$r>0$$ , the two equations determine the quadrant as well as the reference angle. A calculator command based only on $\arctan(y/x)$ can be misleading, because the same ratio occurs in opposite quadrants and because it is undefined when $$x=0$$ . Use a quadrant-aware angle function such as $\operatorname{atan2}(y,x)$ when available, or use the signs of both $$x$$ and $$y$$ to select the correct argument.

Example — rectangular to polar form

For $z=-\sqrt3+i$ , the modulus is

r=\sqrt{(-\sqrt3)^2+1^2}=2.

The reference angle is $\pi/6$ , but the point lies in quadrant II. Therefore a principal argument is $5\pi/6$ , and

z=2\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right).

The value $-\pi/6$ has the same tangent but is in the wrong quadrant, so it is not an argument of this point.

The $$n$$ roots of a complex number form a regular polygon. Solving $$z^n = w$$ produces exactly $$n$$ distinct complex roots evenly distributed on a circle of radius $|w|^{1/n}$ .

Finding the n-th roots

Starting point. Solve $z^n = w = r e^{i\theta}$ . Write $z = R e^{i\phi}$ .

\[R^n e^{in\phi} = r e^{i(\theta + 2\pi k)} \quad (k = 0, 1, \dots, n-1)\]
Account for $2\pi k$ periodicity of angles in the complex plane.reversible
\[R = r^{1/n}, \quad \phi_k = \frac{\theta + 2\pi k}{n}\]
Equate moduli and divide angles by $$n$$ .reversible
\[z_k = r^{1/n} e^{i(\theta + 2\pi k)/n} = r^{1/n} \left( \cos\frac{\theta + 2\pi k}{n} + i\sin\frac{\theta + 2\pi k}{n} \right)\]
List the $$n$$ distinct solutions for $k = 0, 1, \dots, n-1$ .reversible

Conclusion. The $$n$$ roots have equal modulus $r^{1/n}$ and are spaced at equal angle intervals $\Delta\phi = \frac{2\pi}{n}$ , forming the vertices of a regular $$n$$ -gon centred at the origin.

Express $$w$$ in polar form: $w = r e^{i\theta}$ .
Modulus of roots: $R = r^{1/n}$ .
Angles of roots: $\phi_k = \frac{\theta + 2\pi k}{n}$ for $k = 0, 1, 2, \dots, n-1$ .
Convert each $z_k = R(\cos\phi_k + i\sin\phi_k)$ back to Cartesian form if required.

14.4 Polar form and multiplication

Every nonzero complex number can be written as

z=r(\cos\theta+i\sin\theta),\qquad r=|z|\ge0.

Later Euler’s formula writes this more compactly as $z=re^{i\theta}$ . Multiplying polar forms multiplies magnitudes and adds arguments:

r_1e^{i\theta_1}r_2e^{i\theta_2}=r_1r_2e^{i(\theta_1+\theta_2)}.

Geometrically, multiplication performs a scale and a rotation. Division divides magnitudes and subtracts angles.

Geometric picture. In polar form, multiplication combines two independent actions: moduli multiply, so lengths scale; arguments add, so directions rotate. This is why polar multiplication is especially natural for phasors.

r_1e^{i\theta_1}r_2e^{i\theta_2}=r_1r_2e^{i(\theta_1+\theta_2)}

Complex-plane vector before and after scaling and rotation.

14.5 Arithmetic, conjugates, and geometric checks

Complex arithmetic is ordinary polynomial algebra with the one replacement $$i^2=-1$$ . Keeping real and imaginary parts collected is important: a complex number is zero only if both parts are zero. The conjugate changes the sign of the imaginary coordinate, so it is a reflection in the real axis. Its product with the original number is a squared distance:

\[(x+iy)(x-iy)=x^2+y^2=|z|^2.\]

That fact explains division rather than supplying a trick to memorise. A complex denominator is made real by multiplying numerator and denominator by its conjugate; this multiplies the fraction by $$1$$ and changes no value.

Example — multiplication with an error check

Compute $$(2-3i)(4+i)$$ :

\begin{aligned} (2-3i)(4+i) &=8+2i-12i-3i^2\\ &=8-10i+3\\ &=11-10i. \end{aligned}

The final $$+3$$ comes from $$-3i^2=-3(-1)$$ ; losing this sign is one of the most common complex-number errors. As a quick magnitude check,

|2-3i|\,|4+i|=\sqrt{13}\sqrt{17}=\sqrt{221},

while $|11-10i|=\sqrt{121+100}=\sqrt{221}$ , as required.

Example — division from first principles

Write $$(5-i)/(2+3i)$$ in rectangular form:

\frac{5-i}{2+3i} =\frac{(5-i)(2-3i)}{(2+3i)(2-3i)} =\frac{10-15i-2i+3i^2}{4+9}.

Since $$3i^2=-3$$ , this becomes

\frac{7-17i}{13}=\frac7{13}-\frac{17}{13}i.

The denominator is positive and real. If an $$i$$ remains below the fraction bar, the rationalisation step is incomplete.

Multiplication scales moduli and adds arguments. In polar coordinates, complex multiplication decomposes into two independent geometric operations: radial scaling and angular rotation.

Product of two polar complex numbers

Starting point. Let $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$ .

\[z_1 z_2 = (r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = (r_1 r_2) e^{i(\theta_1 + \theta_2)}\]
Use exponent addition rule $e^a e^b = e^{a+b}$ .reversible
\[|z_1 z_2| = |z_1| \cdot |z_2| = r_1 r_2\]
The length of the product is the product of the lengths.reversible
\[\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\]
The angle of the product is the sum of the angles.reversible

Conclusion. $z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$ and $\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$ . Moduli multiply/divide; angles add/subtract.

14.6 Polar form is scale plus rotation

In rectangular coordinates, multiplication requires four products. In polar coordinates, a number has length $$r$$ and direction $\theta$ . Multiplying by $r_2e^{i\theta_2}$ scales every distance by $$r_2$$ and rotates every direction by $\theta_2$ . This gives the rules

|z_1z_2|=|z_1||z_2|, \qquad \arg(z_1z_2)=\arg z_1+\arg z_2\pmod {2\pi}.

For division, $$z_2$$ must be nonzero:

\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}, \qquad \arg\left(\frac{z_1}{z_2}\right)=\arg z_1-\arg z_2\pmod {2\pi}.

[Figure: Complex plane showing

z=re^{i\theta}

, then a second vector after multiplication by

2e^{i\pi/3}

: its length doubled and its angle increased by

\pi/3

Example — powers without repeated rectangular multiplication

Let $z=\sqrt3+i$ . It has modulus $$2$$ and argument $\pi/6$ , so

z^5=2^5e^{i5\pi/6} =32\left(-\frac{\sqrt3}{2}+\frac{i}{2}\right) =-16\sqrt3+16i.

This is De Moivre's formula,

\left[r(\cos\theta+i\sin\theta)\right]^n =r^n(\cos n\theta+i\sin n\theta),

for integer $$n$$ . It follows directly by multiplying magnitudes and adding arguments repeatedly. It is especially efficient for powers and roots, not usually for addition: addition is still simpler in rectangular form.

De Moivre's Theorem powers complex numbers. Repeated multiplication in polar form leads to an elegant formula for integer powers and roots.

De Moivre's Theorem

Starting point. Let $z = r(\cos\theta + i\sin\theta) = r e^{i\theta}$ , and let $$n$$ be an integer.

\[z^n = (r e^{i\theta})^n = r^n e^{in\theta}\]
Apply power rule to the exponential form.reversible
\[z^n = r^n(\cos(n\theta) + i\sin(n\theta))\]
Expand back into trigonometric components using Euler's formula.reversible

Conclusion. $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))$ . Computing $$z^n$$ in polar form takes $$O(1)$$ arithmetic, avoiding tedious binomial expansion.

14.7 Roots and the multivalued argument

To solve $w^n=z=re^{i\theta}$ with $$r>0$$ , the argument of $$z$$ may be $\theta+2\pi k$ for any integer $$k$$ . Dividing by $$n$$ gives the $$n$$ distinct roots

w_k=r^{1/n}e^{i(\theta+2\pi k)/n}, \qquad k=0,1,\ldots,n-1.

Example — cube roots have three distinct directions

Solve $$w^3=-8$$ . Write $-8=8e^{i(\pi+2\pi k)}$ . Its cube roots have modulus $$2$$ and arguments

\frac{\pi}{3},\qquad \pi,\qquad \frac{5\pi}{3}.

Thus the roots are $1+\sqrt3i$ , $$-2$$ , and $1-\sqrt3i$ . They form an equilateral triangle centered at the origin. Reporting only the principal root would miss two algebraic solutions.

Why roots are evenly spaced. Taking an nth root divides an argument by n, but the original angle may differ by any whole turn. Those possible whole turns become angle steps of 2π/n, placing every root on one circle at equal angular intervals.

z_k=r^{1/n}e^{i(\theta+2\pi k)/n},\quad k=0,\ldots,n-1

Equally spaced roots of unity on the complex unit circle.

Common Mistakes

Using $\tan^{-1}(y/x)$ without correcting its quadrant.
Treating $\operatorname{Arg}z$ as defined at $$z=0$$ ; the zero vector has no direction.
Writing $\sqrt{x^2+y^2}=x+y$ or omitting the nonnegative square root.
Adding polar magnitudes and arguments. The simple polar rules are for multiplication and division, not addition.

Exercises

Compute $$(2-3i)(4+i)$$ .
Write $$(1+i)/(1-i)$$ in $$a+ib$$ form.
Find modulus and a principal argument of $-1+\sqrt3i$ .
Convert $4(\cos\frac\pi6+i\sin\frac\pi6)$ to rectangular form.
Explain the geometric effect of multiplying a complex number by $$i$$ .
Compute $$(3+2i)(1-4i)$$ and check the result using moduli.
Write $$(4+3i)/(1-i)$$ in rectangular form.
Express $$(1-i)^8$$ using polar form.
Find all cube roots of $$8i$$ , giving both polar and rectangular forms.