Section CNO  Complex Number Operations

From A First Course in Linear Algebra
Version 2.12
http://linear.ups.edu/

In this section we review 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, but instead illustrate with examples.

Example ACN
Arithmetic of complex numbers

\eqalignno{ (2 + 5i) + (6 − 4i)& = (2 + 6) + (5 + (−4))i = 8 + i && \cr (2 + 5i) − (6 − 4i)& = (2 − 6) + (5 − (−4))i = −4 + 9i && \cr (2 + 5i)(6 − 4i) & = (2)(6) + (5i)(6) + (2)(−4i) + (5i)(−4i) = 12 + 30i − 8i − 20{i}^{2}&& \cr & = 12 + 22i − 20(−1) = 32 + 22i && \text{Division takes just a bit more care. We multiply the denominator by a complex number chosen to produce a real number and then we can produce a complex number as a result.} \cr {2 + 5i\over 6 − 4i} & = {2 + 5i\over 6 − 4i} {6 + 4i\over 6 + 4i} = {−8 + 38i\over 52} = −{8\over 52} + {38\over 52}i = −{2\over 13} + {19\over 26}i &&

}

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 α = a + bi and β = c + di are equal, denoted α = β, if a = c and b = d.

(This definition contains Notation CNE.)

Definition CNA
The sum of the complex numbers α = a + bi and β = c + di , denoted α + β, is (a + c) + (b + d)i.

(This definition contains Notation CNA.)

Definition CNM
Complex Number Multiplication
The product of the complex numbers α = a + bi and β = c + di , denoted αβ, is (ac − bd) + (ad + bc)i.

(This definition contains Notation CNM.)

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 α,β ∈ ℂ, then α + β ∈ ℂ.
• MCCN Multiplicative Closure, Complex Numbers
If α,β ∈ ℂ, then αβ ∈ ℂ.
• CACN Commutativity of Addition, Complex Numbers
For any α,\kern 1.95872pt β ∈ ℂ, α + β = β + α.
• CMCN Commutativity of Multiplication, Complex Numbers
For any α,\kern 1.95872pt β ∈ ℂ, αβ = βα.
• AACN Additive Associativity, Complex Numbers
For any α,\kern 1.95872pt β,\kern 1.95872pt γ ∈ ℂ, α + \left (β + γ\right ) = \left (α + β\right ) + γ.
• MACN Multiplicative Associativity, Complex Numbers
For any α,\kern 1.95872pt β,\kern 1.95872pt γ ∈ ℂ, α\left (βγ\right ) = \left (αβ\right )γ.
• DCN Distributivity, Complex Numbers
For any α,\kern 1.95872pt β,\kern 1.95872pt γ ∈ ℂ, α(β + γ) = αβ + αγ.
• ZCN Zero, Complex Numbers
There is a complex number 0 = 0 + 0i so that for any α ∈ ℂ, 0 + α = α.
• OCN One, Complex Numbers
There is a complex number 1 = 1 + 0i so that for any α ∈ ℂ, 1α = α.
• AICN Additive Inverse, Complex Numbers
For every α ∈ ℂ there exists − α ∈ ℂ so that α + \left (−α\right ) = 0.
• MICN Multiplicative Inverse, Complex Numbers
For every α ∈ ℂ, α\mathrel{≠}0 there exists {1\over α} ∈ ℂ so that α\left ({1\over α}\right ) = 1.

Proof   We could derive each of these properties of complex numbers with a proof that builds on the identical properties of the real numbers. The only proof that might be at all interesting would be to show Property MICN since we would need to trot out a conjugate. For this property, and especially for the others, we might be tempted to construct proofs of the identical properties for the reals. This would take us way too far afield, so we will draw a line in the sand right here and just agree that these nine fundamental behaviors are true. OK?

Mostly we have stated these nine properties carefully so that we can make reference to them later in other proofs. So we will be linking back here often.

Subsection CCN: Conjugates of Complex Numbers

Definition CCN
Conjugate of a Complex Number
The conjugate of the complex number α = a + bi ∈ {ℂ}^{} is the complex number \overline{α} = a − bi.

(This definition contains Notation CCN.)

Example CSCN
Conjugate of some complex numbers

\eqalignno{ \overline{2 + 3i} = 2 − 3i & &\overline{5 − 4i} = 5 + 4i & &\overline{ − 3 + 0i} = −3 + 0i & &\overline{0 + 0i} = 0 + 0i & & & & & & & & }

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
Suppose that α and β are complex numbers. Then \overline{α + β} = \overline{α} + \overline{β}.

Proof   Let α = a + bi and β = r + si. Then

 \overline{α + β} = \overline{(a + r) + (b + s)i} = (a + r) − (b + s)i = (a − bi) + (r − si) = \overline{α} + \overline{β}

Theorem CCRM
Complex Conjugation Respects Multiplication
Suppose that α and β are complex numbers. Then \overline{αβ} = \overline{α}\overline{β}.

Proof   Let α = a + bi and β = r + si. Then

\eqalignno{ \overline{αβ} & = \overline{(ar − bs) + (as + br)i} = (ar − bs) − (as + br)i & & \cr & = (ar − (−b)(−s)) + (a(−s) + (−b)r)i = (a − bi)(r − si) = \overline{α}\overline{β} & & }

Theorem CCT
Complex Conjugation Twice
Suppose that α is a complex number. Then \overline{\overline{α}} = α.

Proof   Let α = a + bi. Then

 \overline{\overline{α}} = \overline{a − bi} = a − (−bi) = a + bi = α

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 α = a + bi ∈ {ℂ}^{}, is the nonnegative real number

 \left \vert α\right \vert = \sqrt{α\overline{α}} = \sqrt{{a}^{2 } + {b}^{2}}.

Example MSCN
Modulus of some complex numbers

\eqalignno{ \left \vert 2 + 3i\right \vert = \sqrt{13} & &\left \vert 5 − 4i\right \vert = \sqrt{41} & &\left \vert −3 + 0i\right \vert = 3 & &\left \vert 0 + 0i\right \vert = 0 & & & & & & & & }

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 \left \vert −3\right \vert = \left \vert −3 + 0i\right \vert = 3. Notice too how the modulus of the complex zero, 0 + 0i, has value 0.