2.2 Complex numbers

Detailed Theory: Complex Numbers

1. Introduction to Complex Numbers

1.1 The Need for Complex Numbers

The equation $x^2 + 1 = 0$ has no real solution because there is no real number whose square is negative. This limitation of real numbers led to the invention of complex numbers.

1.2 Definition of Imaginary Unit

The imaginary unit $i$ is defined as:

$i = \sqrt{-1}$

From this definition, we get:

$i^2 = -1$

1.3 Powers of $i$

The powers of $i$ follow a cyclic pattern with period 4:

$i^1 = i$

$i^2 = -1$

$i^3 = i^2 \cdot i = -i$

$i^4 = i^2 \cdot i^2 = (-1)(-1) = 1$

For any integer $n$, we can compute $i^n$ by dividing $n$ by 4 and using the remainder:

$i^n = i^{4k + r} = i^r$ where $r = 0, 1, 2, 3$

Examples:

$i^{17} = i^{4\cdot4 + 1} = i^1 = i$

$i^{-3} = \frac{1}{i^3} = \frac{1}{-i} = \frac{i}{-i^2} = \frac{i}{-(-1)} = i$

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2. Definition and Representation of Complex Numbers

2.1 Standard Form

A complex number $z$ is expressed in the standard form as:

$z = a + bi$

where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Components:

$a$ is called the real part of $z$, denoted $\text{Re}(z) = a$

$b$ is called the imaginary part of $z$, denoted $\text{Im}(z) = b$

2.2 Set Notation

The set of all complex numbers is denoted by:

$\mathbb{C} = \{a + bi : a, b \in \mathbb{R}\}$

2.3 Equality of Complex Numbers

Two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ are equal if and only if:

$a = c$ and $b = d$

That is, their real parts are equal AND their imaginary parts are equal.

2.4 Special Types of Complex Numbers

a) Purely Real Number

If $b = 0$, then $z = a$ is a purely real number.

Example:

$z = 5$, which is $5 + 0i$

b) Purely Imaginary Number

If $a = 0$, then $z = bi$ is a purely imaginary number.

Example:

$z = 3i$, which is $0 + 3i$

c) Zero Complex Number

If $a = 0$ and $b = 0$, then $z = 0$ is the zero complex number.

d) Complex Conjugate

For $z = a + bi$, its complex conjugate is:

$\overline{z} = a - bi$

Properties:

$\overline{\overline{z}} = z$

If $z$ is purely real, then $\overline{z} = z$

If $z$ is purely imaginary, then $\overline{z} = -z$

2.5 Modulus (Absolute Value)

For $z = a + bi$, the modulus of $z$ is:

$|z| = \sqrt{a^2 + b^2}$

The modulus is always a non-negative real number.

Geometric interpretation: $|z|$ represents the distance from the origin to the point $(a, b)$ in the complex plane.

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3. Geometric Representation (Argand Plane)

3.1 The Complex Plane

The complex plane (Argand plane) is a Cartesian plane where:

The horizontal axis is the real axis

The vertical axis is the imaginary axis

The complex number $z = a + bi$ is represented by the point $(a, b)$

3.2 Polar Form of Complex Numbers

Instead of using rectangular coordinates $(a, b)$, we can represent a complex number using polar coordinates:

$z = r(\cos \theta + i \sin \theta)$

where:

$r = |z| = \sqrt{a^2 + b^2}$ (modulus)

$\theta = \arg(z) = \tan^{-1}\left(\frac{b}{a}\right)$ (argument)

Note: The argument $\theta$ is determined up to multiples of $2\pi$. The principal argument is usually taken in the interval $(-\pi, \pi]$ or $[0, 2\pi)$.

3.3 Euler's Formula

Euler's formula provides an elegant representation:

$e^{i\theta} = \cos \theta + i \sin \theta$

Using this, the polar form becomes:

$z = re^{i\theta}$

This is called the exponential form of a complex number.

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4. Operations on Complex Numbers

4.1 Addition

For $z_1 = a + bi$ and $z_2 = c + di$:

$z_1 + z_2 = (a + c) + (b + d)i$

Geometric interpretation: Vector addition in the complex plane.

Properties:

Commutative: $z_1 + z_2 = z_2 + z_1$

Associative: $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$

Additive identity: $z + 0 = z$

Additive inverse: For $z = a + bi$, $-z = -a - bi$

4.2 Subtraction

For $z_1 = a + bi$ and $z_2 = c + di$:

$z_1 - z_2 = (a - c) + (b - d)i$

4.3 Multiplication

a) Using Standard Form

For $z_1 = a + bi$ and $z_2 = c + di$:

$z_1 \cdot z_2 = (a + bi)(c + di)$

$= ac + adi + bci + bdi^2$

$= ac + (ad + bc)i + bd(-1)$

$= (ac - bd) + (ad + bc)i$

b) Using Polar Form

For $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$:

$z_1 \cdot z_2 = r_1 r_2[\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$

Using exponential form:

$z_1 \cdot z_2 = r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)}$

Geometric interpretation: Multiply the moduli, add the arguments.

Properties:

Commutative: $z_1 \cdot z_2 = z_2 \cdot z_1$

Associative: $(z_1 \cdot z_2) \cdot z_3 = z_1 \cdot (z_2 \cdot z_3)$

Multiplicative identity: $z \cdot 1 = z$

Distributive over addition: $z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3$

4.4 Division

a) Using Standard Form

For $z_1 = a + bi$ and $z_2 = c + di \neq 0$:

$\frac{z_1}{z_2} = \frac{a + bi}{c + di}$

Multiply numerator and denominator by the conjugate of the denominator:

$= \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

$= \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

$= \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} i$

b) Using Polar Form

For $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2) \neq 0$:

$\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$

Using exponential form:

$\frac{z_1}{z_2} = \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$

Geometric interpretation: Divide the moduli, subtract the arguments.

4.5 Properties of Conjugates

For complex numbers $z$, $z_1$, and $z_2$:

1. $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$

2. $\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}$

3. $\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}$

4. $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}$ for $z_2 \neq 0$

5. $z + \overline{z} = 2\text{Re}(z)$

6. $z - \overline{z} = 2i\text{Im}(z)$

7. $z \cdot \overline{z} = |z|^2$

8. $\overline{\overline{z}} = z$

4.6 Properties of Modulus

For complex numbers $z$, $z_1$, and $z_2$:

1. $|z| \geq 0$ and $|z| = 0$ if and only if $z = 0$

2. $|z| = |\overline{z}|$

3. $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$

4. $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ for $z_2 \neq 0$

5. $|z_1 + z_2| \leq |z_1| + |z_2|$ (Triangle Inequality)

6. $||z_1| - |z_2|| \leq |z_1 - z_2|$

7. $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)$ (Parallelogram Law)

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5. De Moivre's Theorem and Applications

5.1 De Moivre's Theorem

For any integer $n$ and any real number $\theta$:

$(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$

Using exponential form:

$(e^{i\theta})^n = e^{in\theta}$

5.2 Finding $n$-th Powers

To compute $(a + bi)^n$:

Step 1: Convert to polar form: $a + bi = r(\cos \theta + i \sin \theta)$

Step 2: Apply De Moivre's theorem:

$(a + bi)^n = r^n[\cos(n\theta) + i \sin(n\theta)]$

Step 3: Convert back to standard form if needed.

Example: Find $(1 + i)^8$

Step 1: Convert to polar form:

$r = \sqrt{1^2 + 1^2} = \sqrt{2}$

$\theta = \frac{\pi}{4}$ (since $a = 1$, $b = 1$)

So $1 + i = \sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$

Step 2: Apply De Moivre:

$(1 + i)^8 = (\sqrt{2})^8[\cos(8 \cdot \frac{\pi}{4}) + i \sin(8 \cdot \frac{\pi}{4})]$

$= 2^4[\cos(2\pi) + i \sin(2\pi)]$

$= 16[1 + 0i] = 16$

5.3 Finding $n$-th Roots

The $n$-th roots of a complex number $z = r(\cos \theta + i \sin \theta)$ are given by:

$z^{1/n} = r^{1/n}\left[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right)\right]$

for $k = 0, 1, 2, \ldots, n-1$

Properties:

1. There are exactly $n$ distinct $n$-th roots of any non-zero complex number.

2. In the complex plane, the $n$-th roots are equally spaced on a circle of radius $r^{1/n}$.

Example: Find the cube roots of $8i$

Step 1: Convert $8i$ to polar form:

$8i = 8(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})$ since $8i = 0 + 8i$

Step 2: Apply formula with $n = 3$:

For $k = 0$: $z_0 = 8^{1/3}[\cos(\frac{\pi/2}{3}) + i \sin(\frac{\pi/2}{3})] = 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})$

$= 2\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) = \sqrt{3} + i$

For $k = 1$: $z_1 = 2[\cos(\frac{\pi/2 + 2\pi}{3}) + i \sin(\frac{\pi/2 + 2\pi}{3})] = 2(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6})$

$= 2\left(-\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) = -\sqrt{3} + i$

For $k = 2$: $z_2 = 2[\cos(\frac{\pi/2 + 4\pi}{3}) + i \sin(\frac{\pi/2 + 4\pi}{3})] = 2(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})$

$= 2(0 - i) = -2i$

The cube roots are: $\sqrt{3} + i$, $-\sqrt{3} + i$, and $-2i$

5.4 Applications of De Moivre's Theorem

a) Expressing $\cos(n\theta)$ and $\sin(n\theta)$

Using the binomial expansion and De Moivre's theorem:

$\cos(n\theta) = \text{Re}[(\cos \theta + i \sin \theta)^n]$

$\sin(n\theta) = \text{Im}[(\cos \theta + i \sin \theta)^n]$

Example: Express $\cos(3\theta)$ in terms of $\cos \theta$

$(\cos \theta + i \sin \theta)^3 = \cos^3 \theta + 3i\cos^2 \theta \sin \theta - 3\cos \theta \sin^2 \theta - i\sin^3 \theta$

By De Moivre: $(\cos \theta + i \sin \theta)^3 = \cos(3\theta) + i \sin(3\theta)$

Equating real parts:

$\cos(3\theta) = \cos^3 \theta - 3\cos \theta \sin^2 \theta$

Using $\sin^2 \theta = 1 - \cos^2 \theta$:

$\cos(3\theta) = \cos^3 \theta - 3\cos \theta (1 - \cos^2 \theta) = 4\cos^3 \theta - 3\cos \theta$

b) Summing Series

De Moivre's theorem can be used to sum trigonometric series using geometric series formula.

Example: Sum $S = \cos \theta + \cos 2\theta + \cdots + \cos n\theta$

Consider $C = \cos \theta + \cos 2\theta + \cdots + \cos n\theta$

and $S = \sin \theta + \sin 2\theta + \cdots + \sin n\theta$

Then $C + iS = e^{i\theta} + e^{i2\theta} + \cdots + e^{in\theta}$

This is a geometric series with first term $e^{i\theta}$, ratio $e^{i\theta}$, and $n$ terms.

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6. Complex Numbers in Quadratic Equations

6.1 Solving Quadratic Equations with Real Coefficients

For a quadratic equation with real coefficients:

$ax^2 + bx + c = 0$ with $a \neq 0$

The discriminant is:

$\Delta = b^2 - 4ac$

Cases:

1. If $\Delta > 0$: Two distinct real roots

2. If $\Delta = 0$: One real root (double root)

3. If $\Delta < 0$: Two complex conjugate roots

6.2 Complex Roots Formula

When $\Delta < 0$, we can write $\Delta = -D$ where $D > 0$

Then the roots are:

$x = \frac{-b \pm i\sqrt{D}}{2a}$

These are complex conjugates of each other.

Example: Solve $x^2 - 2x + 5 = 0$

Here $a=1$, $b=-2$, $c=5$

$\Delta = (-2)^2 - 4(1)(5) = 4 - 20 = -16$

Since $\Delta < 0$, roots are complex:

$x = \frac{2 \pm i\sqrt{16}}{2} = \frac{2 \pm 4i}{2} = 1 \pm 2i$

Roots: $1 + 2i$ and $1 - 2i$ (complex conjugates)

6.3 Sum and Product of Roots

For $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$:

Sum of roots: $\alpha + \beta = -\frac{b}{a}$

Product of roots: $\alpha\beta = \frac{c}{a}$

This holds even when roots are complex.

Verification of previous example:

Sum: $(1+2i) + (1-2i) = 2 = -\frac{-2}{1} \quad \checkmark$

Product: $(1+2i)(1-2i) = 1 - 4i^2 = 1 + 4 = 5 = \frac{5}{1} \quad \checkmark$

6.4 Quadratic with Given Complex Roots

To form a quadratic equation with real coefficients having roots $\alpha$ and $\beta$:

If roots are complex conjugates, use:

$x^2 - (\alpha + \beta)x + \alpha\beta = 0$

Example: Form quadratic with roots $2+3i$ and $2-3i$

Sum: $(2+3i) + (2-3i) = 4$

Product: $(2+3i)(2-3i) = 4 - 9i^2 = 4 + 9 = 13$

Equation: $x^2 - 4x + 13 = 0$

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7. Locus Problems in Complex Plane

7.1 Basic Locus Concepts

A locus in the complex plane is a set of points satisfying certain conditions.

Common conditions:

1. $|z - z_0| = r$: Circle with center $z_0$ and radius $r$

2. $|z - z_1| = |z - z_2|$: Perpendicular bisector of segment joining $z_1$ and $z_2$

3. $\text{Re}(z) = c$: Vertical line through $x = c$

4. $\text{Im}(z) = c$: Horizontal line through $y = c$

5. $\arg(z - z_0) = \theta$: Ray from $z_0$ making angle $\theta$ with positive real axis

7.2 Circle Loci

a) Standard Circle

$|z - z_0| = r$ represents a circle with center at $z_0$ and radius $r$.

If $z_0 = a + bi$ and $z = x + yi$, then:

$|(x + yi) - (a + bi)| = r$

$|(x-a) + (y-b)i| = r$

$(x-a)^2 + (y-b)^2 = r^2$

b) Circle in Different Forms

1. $|z| = r$: Circle centered at origin with radius $r$

2. $|z - i| = 2$: Circle centered at $i$ with radius $2$

3. $|z - 1| = |z + i|$: Set of points equidistant from $1$ and $-i$, which is the perpendicular bisector

7.3 Line Loci

a) Perpendicular Bisector

$|z - z_1| = |z - z_2|$ represents the perpendicular bisector of the segment joining $z_1$ and $z_2$.

Example: $|z - 1| = |z + i|$

Let $z = x + yi$:

$|(x-1) + yi| = |x + (y+1)i|$

$\sqrt{(x-1)^2 + y^2} = \sqrt{x^2 + (y+1)^2}$

Squaring both sides:

$(x-1)^2 + y^2 = x^2 + (y+1)^2$

$x^2 - 2x + 1 + y^2 = x^2 + y^2 + 2y + 1$

$-2x = 2y$

$y = -x$

This is a straight line through origin with slope $-1$.

b) Lines Parallel to Axes

$\text{Re}(z) = k$: Vertical line $x = k$

$\text{Im}(z) = k$: Horizontal line $y = k$

c) Lines at an Angle

$\arg(z - z_0) = \theta$: Ray starting at $z_0$ and making angle $\theta$ with positive real axis.

Example: $\arg(z - 1) = \frac{\pi}{4}$ represents the ray starting at $1$ and making $45^\circ$ angle with positive real axis.

7.4 Ellipse and Hyperbola Loci

a) Ellipse

$|z - z_1| + |z - z_2| = 2a$ (with $2a > |z_1 - z_2|$) represents an ellipse with foci at $z_1$ and $z_2$, and major axis length $2a$.

b) Hyperbola

$||z - z_1| - |z - z_2|| = 2a$ (with $2a < |z_1 - z_2|$) represents a hyperbola with foci at $z_1$ and $z_2$.

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8. Applications and Solved Examples

Example 1: Simplify Complex Expression

Simplify: $\frac{(1+i)^3}{(1-i)^2}$

Solution:

Step 1: Compute $(1+i)^3$:

$1+i$ in polar form: $\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$

By De Moivre: $(1+i)^3 = (\sqrt{2})^3(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}) = 2\sqrt{2}(-\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i)$

$= -2 + 2i$

Step 2: Compute $(1-i)^2$:

$1-i = \sqrt{2}(\cos(-\frac{\pi}{4}) + i \sin(-\frac{\pi}{4}))$

$(1-i)^2 = 2(\cos(-\frac{\pi}{2}) + i \sin(-\frac{\pi}{2})) = 2(0 - i) = -2i$

Step 3: Divide:

$\frac{-2 + 2i}{-2i} = \frac{-2(1 - i)}{-2i} = \frac{1 - i}{i}$

Multiply numerator and denominator by $-i$:

$= \frac{(1-i)(-i)}{i(-i)} = \frac{-i + i^2}{-i^2} = \frac{-i - 1}{1} = -1 - i$

Example 2: Solve Complex Equation

Solve: $z^2 + 2z + 5 = 0$

Solution:

Using quadratic formula:

$z = \frac{-2 \pm \sqrt{4 - 20}}{2} = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i$

Roots: $-1 + 2i$ and $-1 - 2i$

Example 3: Find Locus

Find locus of $z$ satisfying $|z-2| = 3|z+1|$

Solution:

Let $z = x + yi$:

$|(x-2) + yi| = 3|(x+1) + yi|$

$\sqrt{(x-2)^2 + y^2} = 3\sqrt{(x+1)^2 + y^2}$

Square both sides:

$(x-2)^2 + y^2 = 9[(x+1)^2 + y^2]$

$x^2 - 4x + 4 + y^2 = 9(x^2 + 2x + 1 + y^2)$

$x^2 - 4x + 4 + y^2 = 9x^2 + 18x + 9 + 9y^2$

$0 = 8x^2 + 22x + 5 + 8y^2$

Divide by 8:

$x^2 + \frac{11}{4}x + \frac{5}{8} + y^2 = 0$

Complete the square for $x$:

$(x^2 + \frac{11}{4}x + \frac{121}{64}) + y^2 = -\frac{5}{8} + \frac{121}{64}$

$(x + \frac{11}{8})^2 + y^2 = \frac{-40 + 121}{64} = \frac{81}{64}$

$(x + \frac{11}{8})^2 + y^2 = (\frac{9}{8})^2$

This is a circle with center $(-\frac{11}{8}, 0)$ and radius $\frac{9}{8}$

Example 4: Find Modulus and Argument

For $z = 1 - i\sqrt{3}$, find $|z|$ and $\arg(z)$

Solution:

$|z| = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2$

For argument: $\tan \theta = \frac{-\sqrt{3}}{1} = -\sqrt{3}$

Since point is in 4th quadrant (positive real, negative imaginary):

$\theta = -\frac{\pi}{3}$ or $\frac{5\pi}{3}$

Principal argument usually taken in $(-\pi, \pi]$, so $\arg(z) = -\frac{\pi}{3}$

Example 5: Find Cube Roots of Unity

Find the cube roots of 1.

Solution:

Write 1 in polar form: $1 = \cos 0 + i \sin 0$

Cube roots: $1^{1/3} = \cos\left(\frac{0 + 2k\pi}{3}\right) + i \sin\left(\frac{0 + 2k\pi}{3}\right)$ for $k = 0, 1, 2$

For $k=0$: $z_0 = \cos 0 + i \sin 0 = 1$

For $k=1$: $z_1 = \cos\left(\frac{2\pi}{3}\right) + i \sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$

For $k=2$: $z_2 = \cos\left(\frac{4\pi}{3}\right) + i \sin\left(\frac{4\pi}{3}\right) = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$

The cube roots of unity are: $1$, $-\frac{1}{2} + \frac{\sqrt{3}}{2}i$, and $-\frac{1}{2} - \frac{\sqrt{3}}{2}i$

Note: $\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$ is often denoted as the primitive cube root of unity, and satisfies $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$

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9. Important Formulas and Theorems

9.1 Basic Identities

1. $i^2 = -1$

2. $i^3 = -i$

3. $i^4 = 1$

4. $\frac{1}{i} = -i$

5. $(a+bi)(a-bi) = a^2 + b^2$

9.2 Polar Form Relationships

1. $r = |z| = \sqrt{a^2 + b^2}$

2. $\theta = \arg(z) = \tan^{-1}\left(\frac{b}{a}\right)$ (with quadrant consideration)

3. $a = r\cos\theta$

4. $b = r\sin\theta$

5. $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

9.3 De Moivre's Theorem

For any integer $n$:

$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$

9.4 $n$-th Roots Formula

For $z = r(\cos\theta + i\sin\theta) \neq 0$, the $n$-th roots are:

$z_k = r^{1/n}\left[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right]$

for $k = 0, 1, 2, \ldots, n-1$

9.5 Cube Roots of Unity

The cube roots of 1 are:

$1$, $\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$, and $\omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$

Properties:

$\omega^3 = 1$

$1 + \omega + \omega^2 = 0$

$\omega^2 = \overline{\omega}$

9.6 Quadratic Formula with Complex Roots

For $ax^2 + bx + c = 0$ with real coefficients:

If $\Delta = b^2 - 4ac < 0$, then roots are:

$x = \frac{-b \pm i\sqrt{-\Delta}}{2a}$

9.7 Triangle Inequality

For any complex numbers $z_1$ and $z_2$:

$|z_1 + z_2| \leq |z_1| + |z_2|$

$||z_1| - |z_2|| \leq |z_1 - z_2|$

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10. Exam Tips and Common Pitfalls

10.1 Common Mistakes to Avoid

1. Incorrect simplification of powers of $i$: Remember the cyclic nature $i, -1, -i, 1$

2. Forgetting to rationalize denominators: Always write complex numbers in standard form $a+bi$

3. Incorrect argument calculation: Remember to consider the quadrant of the complex number

4. Misapplying De Moivre's theorem: It applies to polar/exponential form, not directly to standard form

5. Ignoring complex conjugate pairs: For polynomials with real coefficients, complex roots occur in conjugate pairs

10.2 Problem-Solving Strategies

1. For computations: Convert to polar form if multiplication/division/powers are involved

2. For equations: Use algebraic manipulation, conjugate properties, or geometric interpretation

3. For locus problems: Substitute $z = x + yi$ and use coordinate geometry

4. For inequalities: Use modulus properties and triangle inequality

5. For root-finding: Use polar form and De Moivre's theorem for $n$-th roots

10.3 Verification Techniques

1. Check conjugates: For real-coefficient polynomials, verify roots are conjugates

2. Check modulus: Use $|z|^2 = z\overline{z}$ to verify computations

3. Check special values: Test $z=0$, $z=1$, $z=i$ when applicable

4. Check dimensions: Ensure real and imaginary parts are correctly identified

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This comprehensive theory covers all aspects of complex numbers with detailed explanations and examples, providing complete preparation for the entrance examination.