Introduction to Complex Numbers
Complex numbers represent a fundamental extension of real numbers, enabling mathematical operations beyond the real number system. Denoted as ℂ, the complex number system finds extensive applications in mathematics, engineering, and physics.
Key Definition
A complex number takes the form \(a + bi\) , where:
- \(a\) : real component
- \(b\) : imaginary component
- \(i\) : imaginary unit where \(i^2 = -1\)
Basic Operations with Complex Numbers
Addition and Subtraction
Addition: \((a+bi)+(c+di)=(a+c)+(b+d)i\)
Subtraction: \((a+bi)-(c+di)=(a-c)+(b-d)i\)
Multiplication and Division
Multiplication: \((a+bi)(c+di)=(ac-bd)+(ad+bc)i\)
Division: \(\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}\)
Properties and Forms
Complex Conjugate
For a complex number \(z=a+bi\) , its conjugate is:
\(\overline{z}=a-bi\)
Modulus and Argument
Modulus: \(|z|=\sqrt{a^2+b^2}\)
Argument: \(\arg(z)=\arctan(\frac{b}{a})\)
Alternative Representations
Polar Form
\(z=r(\cos\theta+i\sin\theta)\)
Exponential Form
\(z=re^{i\theta}\)
Powers and Roots of Complex Numbers
Powers of Complex Numbers
For a complex number in exponential form \(z=re^{i\theta}\) :
\(z^n=(re^{i\theta})^n=r^ne^{in\theta}\)
where n is a positive integer
Complex Roots
The n -th root of a complex number has n distinct values:
\(w_k=r^{\frac{1}{n}}e^{\frac{i(\theta+2k\pi)}{n}}\)
where \(k = 0,1,2,\ldots,n-1\)
Key Properties
- Every complex number (except 0) has exactly n distinct n -th roots
- The roots form a regular polygon in the complex plane
- Each successive root is obtained by rotating through an angle of \(\frac{2\pi}{n}\)
Advanced Applications
De Moivre's Theorem
\((\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)\)
Complex Analysis Foundations
Cauchy-Riemann Equations
For a complex function \(f(z)=u(x,y)+iv(x,y)\):
\(\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\)