Chapter MAA60
Complex Numbers
Advanced Topics
Introduction to Complex Numbers
Complex numbers extend the real number system by introducing the imaginary unit
, where
. This allows us to solve equations like
, which have no real number solutions.
A complex number is represented in the form
, where
and
are real numbers. The real part of
is
, and the imaginary part is
.
Components of a Complex Number
For a complex number
:
(Real part)
(Imaginary part)
Special Cases:
If
, then
is a real number
If
, then
is a pure imaginary number
If
and
, then
Problem: Express
in terms of
.
1
We know that
, so
2
Rewrite
as
3
Apply the property
:
4
Therefore,
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
is plotted as the point
in the complex plane. This geometric interpretation helps us understand operations and properties of complex numbers.
Complex Conjugate
The complex conjugate of
, denoted
or
, is defined as:
Geometrically,
is the reflection of
across the real axis in the complex plane.
The modulus (or absolute value) of a complex number
is the distance from the origin to the point
in the complex plane.
Problem: Find the modulus and complex conjugate of
.
1
For
, we have
and
2
The modulus is:
3
The complex conjugate is:
4
Therefore,
and
Rectangular Form of Complex Numbers
The rectangular form (or Cartesian form) of a complex number is
, where
and
are real numbers. This form is most useful for addition and subtraction of complex numbers.
Properties of Rectangular Form
if and only if
and
Problem: Verify that
for
.
1
For
, the complex conjugate is
2
Compute the product:
3
Compute the squared modulus:
4
Since
, 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
Multiplication
Division
Problem: Calculate
.
1
To divide complex numbers, we multiply both numerator and denominator by the conjugate of the denominator:
2
Simplify the denominator:
3
Expand the numerator:
4
Compute the quotient:
Polar Form of Complex Numbers
The polar form of a complex number expresses it in terms of its modulus and argument. If
, then:
where
is the modulus and
is the argument, given by
with appropriate adjustments for the quadrant.
Converting Between Forms
Rectangular to Polar
Polar to Rectangular
Problem: Convert
to polar form.
1
Calculate the modulus:
2
Find the argument:
Since
and
,
is in the second quadrant:
3
Express
in polar form:
Euler's Formula
Euler's formula provides a profound connection between exponential and trigonometric functions through complex numbers:
This allows us to express the polar form of a complex number more concisely as
.
Consequences of Euler's Formula
(Euler's identity)
Problem: Express
using Euler's formula, and calculate
.
1
Using Euler's formula:
2
To calculate
, we use the property of exponentials:
3
Converting back to trigonometric form:
(rounded)
De Moivre's Theorem
De Moivre's Theorem provides a formula for computing integer powers of complex numbers in polar form:
Using Euler's formula, this can be expressed more concisely as:
De Moivre's Theorem is especially useful for finding powers and roots of complex numbers.
Problem: Use De Moivre's Theorem to calculate
.
1
Convert
to polar form:
2
Express in polar form:
3
Apply De Moivre's Theorem:
4
Convert back to rectangular form:
5
Therefore,
Powers and Roots of Complex Numbers
De Moivre's Theorem can be extended to find the
-th roots of a complex number. If
, then the
distinct values of
are:
where
. These roots form a regular
-gon in the complex plane.
Problem: Find all the cube roots of
.
1
Express
in polar form:
2
Apply the formula for cube roots with
:
where
3
Calculate each root:
For
:
For
:
For
:
4
The three cube roots of
are
,
, and
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
has exactly
roots in the complex number system (counting multiplicities).
For a quadratic equation
, the solutions are:
When
, 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
is a root, then
is also a root.
Vieta's Formulas
For a monic polynomial
with roots
:
Problem: Solve the equation
.
1
Rewrite as
2
We need to find the fourth roots of 1, which can be written as
3
Using the formula for
-th roots with
:
where
4
Calculate each root:
For
:
For
:
For
:
For
:
5
Therefore, the solutions to
are
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
, with
being resistance and
reactance.
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:
Fractals
The Mandelbrot set is defined by iterating
for complex values of
and
.
Problem: In an AC circuit, the impedance is
ohms. If the current is
amperes, find the voltage and the power dissipated.
1
Using Ohm's law,
:
2
Convert
to polar form:
where
radians
3
Calculate the voltage:
4
The power is given by
:
watts
5
Therefore, the voltage is approximately
volts and the power dissipated is 12 watts.