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Introduction to Complex Numbers

Complex numbers extend the real number system by introducing the imaginary unit

i

, where

i^2 = -1

. This allows us to solve equations like

x^2 + 1 = 0

, which have no real number solutions.

A complex number is represented in the form

z = a + bi

, where

a

and

b

are real numbers. The real part of

z

is

a

, and the imaginary part is

b

.

Components of a Complex Number

For a complex number

z = a + bi

:

\text{Re}(z) = a

(Real part)

\text{Im}(z) = b

(Imaginary part)

Special Cases:

If

b = 0

, then

z = a

is a real number

If

a = 0

, then

z = bi

is a pure imaginary number

If

a = 0

and

b = 0

, then

z = 0

Problem: Express

\sqrt{-9}

in terms of

i

.

1

We know that

i^2 = -1

, so

\sqrt{-1} = i

2

Rewrite

\sqrt{-9}

as

\sqrt{9 \cdot (-1)}

3

Apply the property

\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}

:

\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i

4

Therefore,

\sqrt{-9} = 3i

The Complex Plane

Complex numbers can be visualized geometrically in the complex plane, also known as the Argand plane. In this representation, the horizontal axis corresponds to the real part, and the vertical axis corresponds to the imaginary part.

The complex number

z = a + bi

is plotted as the point

(a, b)

in the complex plane. This geometric interpretation helps us understand operations and properties of complex numbers.

Complex Conjugate

The complex conjugate of

z = a + bi

, denoted

\overline{z}

or

z^*

, is defined as:

\overline{z} = a - bi

Geometrically,

\overline{z}

is the reflection of

z

across the real axis in the complex plane.

The modulus (or absolute value) of a complex number

z = a + bi

is the distance from the origin to the point

(a, b)

in the complex plane.

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

Problem: Find the modulus and complex conjugate of

z = 3 - 4i

.

1

For

z = 3 - 4i

, we have

a = 3

and

b = -4

2

The modulus is:

|z| = \sqrt{a^2 + b^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

3

The complex conjugate is:

\overline{z} = a - bi = 3 - (-4)i = 3 + 4i

4

Therefore,

|z| = 5

and

\overline{z} = 3 + 4i

Rectangular Form of Complex Numbers

The rectangular form (or Cartesian form) of a complex number is

z = a + bi

, where

a

and

b

are real numbers. This form is most useful for addition and subtraction of complex numbers.

Properties of Rectangular Form

z_1 = z_2

if and only if

a_1 = a_2

and

b_1 = b_2

z \cdot \overline{z} = |z|^2 = a^2 + b^2

\text{Re}(z) = \frac{z + \overline{z}}{2}

\text{Im}(z) = \frac{z - \overline{z}}{2i}

Problem: Verify that

z \cdot \overline{z} = |z|^2

for

z = 2 + 3i

.

1

For

z = 2 + 3i

, the complex conjugate is

\overline{z} = 2 - 3i

2

Compute the product:

z \cdot \overline{z} = (2 + 3i)(2 - 3i) = 4 - 6i + 6i - 9i^2 = 4 + 9 = 13

3

Compute the squared modulus:

|z|^2 = 2^2 + 3^2 = 4 + 9 = 13

4

Since

z \cdot \overline{z} = |z|^2 = 13

, the property is verified.

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided following specific rules.

Basic Operations

Addition and Subtraction

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i

Multiplication

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Division

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

Problem: Calculate

(3 + 2i) \div (1 - i)

.

1

To divide complex numbers, we multiply both numerator and denominator by the conjugate of the denominator:

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

2

Simplify the denominator:

(1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 2

3

Expand the numerator:

(3 + 2i)(1 + i) = 3 + 3i + 2i + 2i^2 = 3 + 5i + 2(-1) = 3 - 2 + 5i = 1 + 5i

4

Compute the quotient:

\frac{3 + 2i}{1 - i} = \frac{1 + 5i}{2} = \frac{1}{2} + \frac{5}{2}i

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its modulus and argument. If

z = a + bi

, then:

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

where

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

is the modulus and

\theta = \arg(z)

is the argument, given by

\theta = \tan^{-1}(b/a)

with appropriate adjustments for the quadrant.

Converting Between Forms

Rectangular to Polar

r = \sqrt{a^2 + b^2}

\theta = \begin{cases} \tan^{-1}(b/a) & \text{if } a > 0 \\ \tan^{-1}(b/a) + \pi & \text{if } a < 0 \\ \pi/2 & \text{if } a = 0, b > 0 \\ -\pi/2 & \text{if } a = 0, b < 0 \\ \text{undefined} & \text{if } a = 0, b = 0 \end{cases}

Polar to Rectangular

a = r\cos\theta

b = r\sin\theta

Problem: Convert

z = -2 + 2i

to polar form.

1

Calculate the modulus:

r = |z| = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}

2

Find the argument:

\tan(\theta) = \frac{b}{a} = \frac{2}{-2} = -1

Since

a < 0

and

b > 0

,

\theta

is in the second quadrant:

\theta = \tan^{-1}(-1) + \pi = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}

3

Express

z

in polar form:

z = 2\sqrt{2}\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right)

Euler's Formula

Euler's formula provides a profound connection between exponential and trigonometric functions through complex numbers:

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

This allows us to express the polar form of a complex number more concisely as

z = re^{i\theta}

.

Consequences of Euler's Formula

e^{i\pi} + 1 = 0

(Euler's identity)

\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}

\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}

Problem: Express

z = 3(\cos(\pi/6) + i\sin(\pi/6))

using Euler's formula, and calculate

z^4

.

1

Using Euler's formula:

z = 3(\cos(\pi/6) + i\sin(\pi/6)) = 3e^{i\pi/6}

2

To calculate

z^4

, we use the property of exponentials:

z^4 = (3e^{i\pi/6})^4 = 3^4 \cdot e^{i\pi/6 \cdot 4} = 81e^{2\pi i/3}

3

Converting back to trigonometric form:

z^4 = 81(\cos(2\pi/3) + i\sin(2\pi/3)) = 81(-1/2 + i\sqrt{3}/2) = -40.5 + 70.2i

(rounded)

De Moivre's Theorem

De Moivre's Theorem provides a formula for computing integer powers of complex numbers in polar form:

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

Using Euler's formula, this can be expressed more concisely as:

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

De Moivre's Theorem is especially useful for finding powers and roots of complex numbers.

Problem: Use De Moivre's Theorem to calculate

(1 + i)^6

.

1

Convert

1 + i

to polar form:

r = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}

\theta = \tan^{-1}(1/1) = \pi/4

2

Express in polar form:

1 + i = \sqrt{2}(\cos(\pi/4) + i\sin(\pi/4)) = \sqrt{2}e^{i\pi/4}

3

Apply De Moivre's Theorem:

(1 + i)^6 = (\sqrt{2}e^{i\pi/4})^6 = (\sqrt{2})^6 \cdot e^{i\pi/4 \cdot 6} = 8 \cdot e^{i3\pi/2}

4

Convert back to rectangular form:

8e^{i3\pi/2} = 8(\cos(3\pi/2) + i\sin(3\pi/2)) = 8(0 - i) = -8i

5

Therefore,

(1 + i)^6 = -8i

Powers and Roots of Complex Numbers

De Moivre's Theorem can be extended to find the

n

-th roots of a complex number. If

z = re^{i\theta}

, then the

n

distinct values of

z^{1/n}

are:

z^{1/n} = r^{1/n}e^{i(\theta + 2\pi k)/n}

where

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

. These roots form a regular

n

-gon in the complex plane.

Problem: Find all the cube roots of

8i

.

1

Express

8i

in polar form:

r = |8i| = 8

\theta = \arg(8i) = \pi/2

8i = 8e^{i\pi/2}

2

Apply the formula for cube roots with

n = 3

:

(8i)^{1/3} = 8^{1/3}e^{i(\pi/2 + 2\pi k)/3} = 2e^{i(\pi/2 + 2\pi k)/3}

where

k = 0, 1, 2

3

Calculate each root:

For

k = 0

:

z_0 = 2e^{i\pi/6} = 2(\cos(\pi/6) + i\sin(\pi/6)) = 2(\sqrt{3}/2 + i/2) = \sqrt{3} + i

For

k = 1

:

z_1 = 2e^{i5\pi/6} = 2(\cos(5\pi/6) + i\sin(5\pi/6)) = 2(-\sqrt{3}/2 + i/2) = -\sqrt{3} + i

For

k = 2

:

z_2 = 2e^{i3\pi/2} = 2(\cos(3\pi/2) + i\sin(3\pi/2)) = 2(0 - i) = -2i

4

The three cube roots of

8i

are

\sqrt{3} + i

,

-\sqrt{3} + i

, and

-2i

Solving Polynomial Equations

One of the most significant applications of complex numbers is in solving polynomial equations. The Fundamental Theorem of Algebra states that every polynomial equation of degree

n \geq 1

has exactly

n

roots in the complex number system (counting multiplicities).

For a quadratic equation

ax^2 + bx + c = 0

, the solutions are:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

When

b^2 - 4ac < 0

, the solutions are complex conjugates.

Complex Roots of Polynomials with Real Coefficients

If a polynomial has real coefficients, then complex roots always occur in conjugate pairs. That is, if

a + bi

is a root, then

a - bi

is also a root.

Vieta's Formulas

For a monic polynomial

x^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0

with roots

r_1, r_2, \ldots, r_n

:

r_1 + r_2 + \ldots + r_n = -a_{n-1}

r_1r_2 + r_1r_3 + \ldots + r_{n-1}r_n = a_{n-2}

\ldots

r_1r_2\ldots r_n = (-1)^n a_0

Problem: Solve the equation

x^4 - 1 = 0

.

1

Rewrite as

x^4 = 1

2

We need to find the fourth roots of 1, which can be written as

1 = 1e^{i0}

3

Using the formula for

n

-th roots with

n = 4

:

1^{1/4} = 1^{1/4}e^{i(0 + 2\pi k)/4} = e^{i\pi k/2}

where

k = 0, 1, 2, 3

4

Calculate each root:

For

k = 0

:

z_0 = e^{i\cdot 0} = 1

For

k = 1

:

z_1 = e^{i\pi/2} = i

For

k = 2

:

z_2 = e^{i\pi} = -1

For

k = 3

:

z_3 = e^{i3\pi/2} = -i

5

Therefore, the solutions to

x^4 - 1 = 0

are

x = 1, i, -1, -i

Applications of Complex Numbers

Complex numbers have numerous applications across various fields of mathematics, science, and engineering.

Electrical Engineering

Complex numbers are used to analyze AC circuits, where impedance is represented as

Z = R + jX

, with

R

being resistance and

X

reactance.

Z = |Z|e^{j\phi}

Control Systems

The poles and zeros of transfer functions in the complex plane determine system stability and response.

Quantum Mechanics

Wave functions in quantum mechanics involve complex numbers, where the probability amplitude is given by:

P(x) = |\Psi(x)|^2 = \Psi^*(x)\Psi(x)

Fractals

The Mandelbrot set is defined by iterating

z_{n+1} = z_n^2 + c

for complex values of

z

and

c

.

Problem: In an AC circuit, the impedance is

Z = 3 + 4i

ohms. If the current is

I = 2e^{i\pi/3}

amperes, find the voltage and the power dissipated.

1

Using Ohm's law,

V = IZ

:

V = 2e^{i\pi/3} \cdot (3 + 4i)

2

Convert

3 + 4i

to polar form:

3 + 4i = 5e^{i\tan^{-1}(4/3)} = 5e^{i\theta}

where

\theta \approx 0.9273

radians

3

Calculate the voltage:

V = 2e^{i\pi/3} \cdot 5e^{i\theta} = 10e^{i(\pi/3 + \theta)} \approx 10e^{i1.9807}

4

The power is given by

P = |I|^2 \cdot \text{Re}(Z)

:

P = |2|^2 \cdot 3 = 4 \cdot 3 = 12

watts

5

Therefore, the voltage is approximately

10e^{i1.9807}

volts and the power dissipated is 12 watts.

Sequences and Series

Exponents and Logarithms

Functions

Polynomial Functions

Rational Functions

Exponential and Log Functions

Trigonometric Functions

Trigonometric Identities

Solving Triangles

Vectors

Limits and Continuity

Derivatives

Integrals

Descriptive Statistics

Probability Concepts

Probability Distributions

Complex Numbers

Further Calculus

Matrices

Applications and Modeling

IA Preparation (Math)

IA Preparation (Physics)

Complex Numbers

Introduction to Complex Numbers

Components of a Complex Number

The Complex Plane

Rectangular Form of Complex Numbers

Properties of Rectangular Form

Operations with Complex Numbers

Basic Operations

Polar Form of Complex Numbers

Converting Between Forms

Euler's Formula

De Moivre's Theorem

Powers and Roots of Complex Numbers

Solving Polynomial Equations

Applications of Complex Numbers