Complex Number

Three ways to represent a complex number.

Standard Form

Definition: A complex number in standard form is an expression of the form where and satisfies . The set of all complex numbers is denoted by


Note that I learned to use rather than since that letter is used in the modelling of electric networks.


is NOT equal to . If it were, then

Real and Imaginary Parts

Let with .

Inverse of Complex Number

For , , exists if and only if . Besides, when it exists,


The complex conjugate of with is

Properties of Conjugates. Let with where . Then


The modulus of with is the nonnegative real number


The modulus is basically the absolute value if we are dealing with real numbers, and this is actually where the modulus comes from.

Properties of Modulus. Let . Then

Geometry of Graphically

Polar Form

There is another way we can represent imaginary numbers, which will be super useful for complex multiplication and division.

We invent this plot, where the x-axis are the Real numbers and y-axis are the imaginary numbers. We see from the above image that

The polar form of a complex number is given by where and is an argument of .

Polar Form: Use z=r\angle\theta

When I first learned polar form in MATH115, we always wrote it as , but this is super long. Instead, just use which provides the same amount of information in a very concise way.

Exercise: Write in polar form.

Since we have and , we know that Answer:

Polar Multiplication and Division

Multiplication Division

The MATH115 notation…

Powers of Complex Numbers

de Moivre’s Theorem. If , then for any .

Complex nth Roots

Let be nonzero, and let n be a positive integer. Then the n distinct nth roots of z are given by for

Complex Exponential Form

Why is this form useful?

Exponential form is really cool, because we can actually start relating exponentials with sins and angles!

Derived from Euler Formula, the complex exponential form of is given by

Complex Polynomials

Theorem 5.2. Let be a real polynomial. If is a root of , then so too is .