Section CNO Complex Number Operations

In this section we review some of the basics of working with complex numbers.

Subsection CNA Arithmetic with complex numbers

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.

Example ACN Arithmetic of complex numbers

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.

Definition CNE Complex Number Equality

The complex numbers $\alpha=a+bi$ and $\beta=c+di$ are equal, denoted $\alpha=\beta$, if $a=c$ and $b=d$.

Definition CNA Complex Number Addition

The sum of the complex numbers $\alpha=a+bi$ and $\beta=c+di$ , denoted $\alpha+\beta$, is $(a+c)+(b+d)i$.

Definition CNM Complex Number Multiplication

The product of the complex numbers $\alpha=a+bi$ and $\beta=c+di$ , denoted $\alpha\beta$, is $(ac-bd)+(ad+bc)i$.

Theorem PCNA Properties of Complex Number Arithmetic

The operations of addition and multiplication of complex numbers have the following properties.

  • ACCN Additive Closure, Complex Numbers
    If $\alpha,\beta\in\complexes$, then $\alpha+\beta\in\complexes$.

  • MCCN Multiplicative Closure, Complex Numbers
    If $\alpha,\beta\in\complexes$, then $\alpha\beta\in\complexes$.

  • CACN Commutativity of Addition, Complex Numbers
    For any $\alpha,\,\beta\in\complexes$, $\alpha+\beta=\beta+\alpha$.

  • CMCN Commutativity of Multiplication, Complex Numbers
    For any $\alpha,\,\beta\in\complexes$, $\alpha\beta=\beta\alpha$.

  • AACN Additive Associativity, Complex Numbers
    For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha+\left(\beta+\gamma\right)=\left(\alpha+\beta\right)+\gamma$.

  • MACN Multiplicative Associativity, Complex Numbers
    For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha\left(\beta\gamma\right)=\left(\alpha\beta\right)\gamma$.

  • DCN Distributivity, Complex Numbers
    For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma$.

  • ZCN Zero, Complex Numbers
    There is a complex number $0=0+0i$ so that for any $\alpha\in\complexes$, $0+\alpha=\alpha$.

  • OCN One, Complex Numbers
    There is a complex number $1=1+0i$ so that for any $\alpha\in\complexes$, $1\alpha=\alpha$.

  • AICN Additive Inverse, Complex Numbers
    For every $\alpha\in\complexes$ there exists $-\alpha\in\complexes$ so that $\alpha+\left(-\alpha\right)=0$.

  • MICN Multiplicative Inverse, Complex Numbers
    For every $\alpha\in\complexes$, $\alpha\neq 0$ there exists $\frac{1}{\alpha}\in\complexes$ so that $\alpha\left(\frac{1}{\alpha}\right)=1$.

Proof

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, Theorem ZSSM and Theorem ZVSM.

Theorem ZPCN Zero Product, Complex Numbers

Suppose $\alpha\in\complexes$. Then $0\alpha=0$.

Proof

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, Theorem SMEZV.

Theorem ZPZT Zero Product, Zero Terms

Suppose $\alpha,\beta\in\complexes$. Then $\alpha\beta=0$ if and only if at least one of $\alpha=0$ or $\beta=0$.

Proof

As an equivalence (Proof Technique E), we could restate this result as the contrapositive (Proof Technique CP) by negating each statement, so it would read “$\alpha\beta\neq 0$ if and only if $\alpha\neq 0$ and $\beta\neq 0$.” After you have learned more about nonsingular matrices and matrix multiplication, you should compare this result with Theorem NPNT.

Subsection CCN Conjugates of Complex Numbers

Definition CCN Conjugate of a Complex Number

The conjugate of the complex number $\alpha=a+bi\in\complex{\null}$ is the complex number $\conjugate{\alpha}=a-bi$.

Example CSCN Conjugate of some complex numbers

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.

Theorem CCRA Complex Conjugation Respects Addition

Suppose that $\alpha$ and $\beta$ are complex numbers. Then $\conjugate{\alpha+\beta}=\conjugate{\alpha}+\conjugate{\beta}$.

Proof
Theorem CCRM Complex Conjugation Respects Multiplication

Suppose that $\alpha$ and $\beta$ are complex numbers. Then $\conjugate{\alpha\beta}=\conjugate{\alpha}\conjugate{\beta}$.

Proof
Theorem CCT Complex Conjugation Twice

Suppose that $\alpha$ is a complex number. Then $\conjugate{\conjugate{\alpha}}=\alpha$.

Proof

Subsection MCN Modulus of a Complex Number

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

Definition MCN Modulus of a Complex Number

The modulus of the complex number $\alpha=a+bi\in\complex{\null}$, is the nonnegative real number \begin{equation*} \modulus{\alpha}=\sqrt{\conjugate{\alpha}\alpha}=\sqrt{a^2+b^2}. \end{equation*}

Example MSCN Modulus of some complex numbers

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 $0$.