Complex Number Operations

A complex number is a linear combination of $1$ and $i=\sqrt{-1}$, typically written in the form $a+bi$. Complex numbers can be added, subtracted, multiplied and divided, just like we are used to doing with real numbers, including the restriction on division by zero. We will not define these operations carefully immediately, but instead first illustrate with examples.

In this example, we used $6+4i$ to convert the denominator in the fraction to a real number. This number is known as the conjugate, which we define in the next section.

We will often exploit the basic properties of complex number addition, subtraction, multiplication and division, so we will carefully define the two basic operations, together with a definition of equality, and then collect nine basic properties in a theorem.

Zero and one play special roles, of course, and especially zero. Our first result is one we take for granted, but it requires a proof, derived from our nine properties. You can compare it to its vector space counterparts, and .

Our next theorem could be called cancellation, since it will make that possible. Though you will never see us drawing slashes through parts of products. We will also make very limited use of this result, or its vector space counterpart, .

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

Notice how the conjugate of a real number leaves the number unchanged. The conjugate enjoys some basic properties that are useful when we work with linear expressions involving addition and multiplication.

We define one more operation with complex numbers that may be new to you.

The modulus can be interpreted as a version of the absolute value for complex numbers, as is suggested by the notation employed. You can see this in how $\modulus{-3}=\modulus{-3+0i}=3$. Notice too how the modulus of the complex zero, $0+0i$, has value $0$.