Complex Numbers

Introduction

Complex numbers extend the real numbers by including a solution to x2=−1. This equation has no real solution, but by introducing the imaginary unit i with i2=−1, we create a richer number system in which every polynomial equation has solutions.

Complex numbers appear throughout mathematics, physics, and engineering. They simplify trigonometric identities, describe oscillations and waves, enable elegant solutions in differential equations, and are essential in quantum mechanics and electrical engineering.

Basic Definitions

Standard Form

A complex number has the form z=a+b*i, where a,b∈R. We call a=Re(z) the real part and b=Im(z) the imaginary part.

The set of all complex numbers is denoted C. Real numbers are the special case where b=0, and purely imaginary numbers have a=0.

Complex Conjugate

The complex conjugate of z=a+b*i is z=a−b*i. Geometrically, this reflects z across the real axis.

Key properties:

((z_1)+(z_2))=(z_1)+(z_2),((z_1)*(z_2))=(z_1)*(z_2),z*z=a2+b2.

The product z*z is always real and nonnegative.

Modulus

The modulus (absolute value) of z=a+b*i is |z|=√(,a2+b2)=√(,z*z).

Geometrically, |z| is the distance from z to the origin in the complex plane.

Arithmetic Operations

Addition and Subtraction

Add real and imaginary parts separately:

(a+b*i)+(c+d(i))=(a+c)+(b+d)*i.

Multiplication

Use distributivity and i2=−1:

(a+b*i)*(c+d(i))=(a*c−b*d)+(a*d+b*c)*i.

Division

Multiply numerator and denominator by the conjugate of the denominator:

(a+b*i)/(c+d(i))=((a+b*i)*(c−d(i)))/((c+d(i))*(c−d(i)))=((a*c+b*d)+(b*c−a*d)*i)/(c2+d2).

The Complex Plane

Complex numbers can be visualized as points in the complex (Argand) plane. The number z=a+b*i corresponds to the point (a,b).

The horizontal axis is the real axis; the vertical axis is the imaginary axis. Addition corresponds to vector addition in this plane.

Polar Form

Any complex number can be written in polar form using its modulus r=|z| and argument θ=arg(z):

z=r*(cos(θ)+i*sin(θ))=r*cis*θ.

Here r=√(,a2+b2), and θ=arctan(b/a) (adjusted for quadrant).

The argument is defined up to multiples of 2*π; the principal value is usually taken in (−π,π].

Euler's Formula

Euler’s formula connects exponentials and trigonometry:

e(i*θ)=cos(θ)+i*sin(θ).

Thus polar form can be written compactly as z=r*e(i*θ).

The special case θ=π gives Euler’s identity:

e(i*π)+1=0.

Multiplication and Division in Polar Form

Multiplication:

(r_1)*e(i*(θ_1))⋅(r_2)*e(i*(θ_2))=(r_1)*(r_2)*e(i*((θ_1)+(θ_2))).

Division:

((r_1)*e(i*(θ_1)))/((r_2)*e(i*(θ_2)))=(r_1)/(r_2)*e(i*((θ_1)−(θ_2))).

De Moivre's Theorem

For any integer n:

(cos(θ)+i*sin(θ))n=cos(n*θ)+i*sin(n*θ).

In exponential form:

(e(i*θ))n=e(i*n*θ).

Roots of Complex Numbers

The nth roots of z=r*e(i*θ) are:

z1/n=r1/n*e(i*(θ+2*π*k))/n,k=0,1,…,n−1.

There are exactly n distinct roots, equally spaced around a circle of radius r(1/n).

Roots of Unity

Solutions to zn=1:

(ω_k)=e(2*π*i*k)/n=cos((2*π*k)/n)+i*sin((2*π*k)/n).

They form a regular n-gon on the unit circle.

Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has at least one complex root. Therefore, every polynomial of degree n factors completely into n linear factors over C.

The complex numbers are algebraically closed.

Applications

Complex exponentials simplify oscillatory phenomena. A wave

A*cos(ω*t+ϕ)=Re*(A*e(i*(ω*t+ϕ)))

Electrical engineering uses complex impedance. Quantum mechanics uses complex wave functions. Complex analysis extends calculus to complex-valued functions.

Summary

Complex numbers z=a+b*i extend the reals by including i2=−1. The conjugate z=a−b*i and modulus |z|=√(,a2+b2) are fundamental.

Polar form z=r*e(i*θ) and Euler’s formula e(i*θ)=cos(θ)+i*sin(θ) simplify multiplication, division, and powers.

Every polynomial of degree n has exactly n complex roots (counting multiplicity). The nth roots of any complex number form a regular n-gon in the complex plane.

Complex Conjugate Root Theorem

If a polynomial has real coefficients and z=a+b*i is a root, then so is z=a−b*i. Complex roots of real polynomials occur in conjugate pairs.

Triangle Inequality

For any complex numbers (z_1) and (z_2):

|(z_1)+(z_2)|≤|(z_1)|+|(z_2)|.

Equality holds when (z_1) and (z_2) have the same argument.