๐ CHAPTER 9: COMPLEX NUMBERS
๐ CHAPTER 9: COMPLEX NUMBERS
๐ CHAPTER 9: COMPLEX NUMBERS
(Basic to Pro Level with Solved Mathematics)
๐ท 9.1 Introduction
Real numbers were once thought to be sufficient for solving all equations. However, equations like
$$ x^2 + 1 = 0 $$
have no real solution. To solve such equations, mathematicians introduced Complex Numbers.
Applications:
- Algebra
- Trigonometry
- Calculus
- Electrical Engineering
- Quantum Mechanics
๐ท 9.2 Imaginary Unit
The imaginary unit is denoted by i:
$$ i = \sqrt{-1} $$
Powers of i:
$$ i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1 $$
Powers of i repeat every 4.
๐ท 9.3 Definition of a Complex Number
A complex number is of the form:
$$ z = a + ib $$
where
a = real part โ Re(z)
b = imaginary part โ Im(z)
Example:
$$ z = 3 + 5i $$
Re(z) = 3,โIm(z) = 5
๐ท 9.4 Equality of Complex Numbers
Two complex numbers are equal iff their real and imaginary parts are equal.
If $$ a + ib = c + id $$ then $$ a = c \quad \text{and} \quad b = d $$
๐ท 9.5 Algebraic Operations
๐น (A) Addition
$$ (a+ib) + (c+id) = (a+c) + i(b+d) $$
๐น (B) Subtraction
$$ (a+ib) – (c+id) = (a-c) + i(b-d) $$
๐น (C) Multiplication
$$ (a+ib)(c+id) = (ac-bd) + i(ad+bc) $$
๐น (D) Division
Multiply numerator and denominator by the conjugate of the denominator.
๐ท 9.6 Complex Conjugate
If $$ z = a + ib $$ then $$ \bar{z} = a – ib $$
Properties:
- $$ z\bar{z} = a^2 + b^2 $$
- $$ \overline{z_1z_2} = \bar{z_1}\bar{z_2} $$
- $$ z + \bar{z} = 2a $$
๐ท 9.7 Modulus of a Complex Number
If $$ z = a + ib $$ then
$$ |z| = \sqrt{a^2 + b^2} $$
Represents distance from origin in the Argand plane.
๐ท 9.8 Argand Plane
- X-axis โ Real axis
- Y-axis โ Imaginary axis
A complex number \( a+ib \) is represented by the point \( (a,b) \).
๐ท 9.9 Polar Form
Let $$ z = a + ib $$
$$ r = |z| = \sqrt{a^2 + b^2} $$ $$ \theta = \tan^{-1}\left(\frac{b}{a}\right) $$
$$ z = r(\cos\theta + i\sin\theta) $$
๐ท 9.10 Eulerโs Form
$$ e^{i\theta} = \cos\theta + i\sin\theta $$
$$ z = re^{i\theta} $$
๐ท 9.11 De Moivreโs Theorem
If $$ z = r(\cos\theta + i\sin\theta) $$ then
$$ z^n = r^n(\cos n\theta + i\sin n\theta) $$
๐น Solved Example
Find \( (1+i)^4 \)
$$ 1+i = \sqrt{2}\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right) $$ $$ (1+i)^4 = (\sqrt{2})^4(\cos\pi+i\sin\pi) $$ $$ = 4(-1) = -4 $$
๐ท 9.12 Roots of a Complex Number
The nth roots of $$ z = r(\cos\theta + i\sin\theta) $$ are:
$$ z_k = r^{1/n}\left[ \cos\left(\frac{\theta+2k\pi}{n}\right) + i\sin\left(\frac{\theta+2k\pi}{n}\right) \right] $$
where \( k = 0,1,2,\dots,n-1 \)
๐ท 9.13 Quadratic Equation
$$ x^2 + 4 = 0 $$ $$ x^2 = -4 \Rightarrow x = \pm 2i $$
๐ท 9.14 Important Properties
- $$ |z_1z_2| = |z_1||z_2| $$
- $$ \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} $$
- Triangle inequality: $$ |z_1+z_2| \le |z_1|+|z_2| $$
๐ท 9.15 Locus Problem
$$ |z-3| = 2 $$
Represents a circle with center \( (3,0) \) and radius 2.
๐ท 9.16 Solved Pro-Level Example
$$ |x+i| = |2x-i| $$
$$ \sqrt{x^2+1} = \sqrt{4x^2+1} $$ $$ x^2+1 = 4x^2+1 $$ $$ 3x^2=0 \Rightarrow x=0 $$
๐ท 9.17 Exam Tips
- Memorize powers of \( i \)
- Use conjugate for division
- Draw Argand plane for locus
- De Moivreโs theorem is high-scoring
๐ท 9.18 Chapter Summary
- Complex numbers extend real numbers
- Algebraic & polar forms are essential
- Modulus gives distance
- De Moivre simplifies powers & roots
๐ฏ Final Note
Complex Numbers are not โcomplexโ โ they are powerful tools that simplify mathematics when understood correctly.
Written By
Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar
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