1.1 Complex Numbers
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We are familiar with the set of real numbers. In earlier courses covering real analysis we learned about convergence of sequences of real numbers, continuity of functions defined on sets of real numbers, and the calculus of functions defined on subintervals of the real numbers.
This course is about the analysis of a different number system: the system of complex numbers. When we learn about limits and continuity in the complex number system, the material will seem for the most part similar to what happens over the real number system. We will spend most of the course learning the calculus of the complex number system, which is distinct and powerful.
We begin the course with what complex numbers are and how we can picture them.
Definition (Complex Number)
A complex number is an expression of the form where and are real numbers. We write for the set of all complex numbers.
We usually use letters like and to represent complex numbers.
Definition (Real and Imaginary Parts)
If then is the real part of and is the imaginary part of . Write for the real part of and write for the imaginary part of .
Both the real and imaginary parts of a complex number are real!
We have defined a complex number to be an expression where and are real. Strictly speaking is merely a formal expression representing two real numbers and together. No meaning has yet been ascribed to the symbols and in the expression.
Thinking of as two real numbers and suggests a geometric representation of as the point in the plane. Representing complex numbers in this way came about in the 18th century, two hundred years after complex numbers first appeared.
Definition (Real and Imaginary Numbers)
We say that is real if and we say that is imaginary if .
In the complex plane, the set of real numbers lie on the horizontal axis (which we will often call the real axis) and the set of imaginary numbers lie on the vertical axis (which we will often call the imaginary axis).
If we picture complex numbers as lying in the plane we can consider Euclidean properties of complex numbers. In particular, every non-zero complex is a certain distance from the origin that we can calculate using the Pythagoras theorem, and every non-zero complex makes an angle with the positive real axis that we can relate to other properties via trigonometry.
Definition (Modulus/Absolute Value)
The modulus or absolute value of the complex number is the quantity
measuring the distance between and the origin.
Example
What is the modulus of ?
Solution:
We calculate that .
We conclude this section with the polar representaton of a complex number. When we plot as a point in the plane the quantity is its distance from the origin. The angle from the positive real axis anti-clockwire to is called an argument of .
We say "an argument" because it is not uniquely determined: if we replace by for any then we describe the same complex number. However, there is always a unique argument .
Definition (Principle Argument)
The principle argument of a non-zero complex number is the value representing the anti-clockwise angle between and the positive real axis.
Although every non-zero complex number has many arguments, each non-zero complex number has only one principle argument.
Definition (Polar Representation)
Fix non-zero. Writing and we call
the polar representation of .
Example
Write the polar representation of .
Solution:
The modulus of is .
There is an angle of between and the positive real axis. This is not an argument of as it a clockwise measurement. is an argument of but not its principle argument as it is not in the range . The principle argument of is and
is the polar representation of .