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.
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}.
Theorem CCRM Complex Conjugation Respects Multiplication
Suppose that \alpha and \beta are complex numbers. Then \conjugate{\alpha\beta}=\conjugate{\alpha}\conjugate{\beta}.
Theorem CCT Complex Conjugation Twice
Suppose that \alpha is a complex number. Then \conjugate{\conjugate{\alpha}}=\alpha.
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.