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Wize University Linear Algebra Textbook > Complex Numbers

Basics of Complex Numbers

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Basics of Complex Numbers

We define ii to be the imaginary unit, an imaginary number such that i2=−1i^2=−1.

Complex Numbers in Standard Form

Complex numbers in standard form (or Cartesian form) are numbers of the form a+bia+bi where a, b∈Ra,\ b\in\mathbb{R}.
The set of complex numbers is denoted C\mathbb{C}.
In the standard form z=a+bi ∈Cz=a+bi\ \in\mathbb{C}:
  • aa is called the real part of zz, denoted: Re(z)=a\text{Re}(z)=a
  • bb is called the imaginary part of zz, denoted: Im(z)=b\text{Im}(z)=b
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Complex Plane

The set of real numbers R\mathbb{R} is a subset of C\mathbb{C} (every real number is a complex number whose imaginary part is 00).

Real numbers: 1D objects represented on the number line. We can think of real numbers as either:
  • a point on the line
  • a vector: an arrow from the origin to the point, indicating size (distance from 0) and direction (positive/negative)

Complex numbers: 2D objects represented on the complex plane. We can think of complex numbers as either:
  • a point on the plane
  • a vector: an arrow from the origin to the point, indicating size and direction.

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Operations With Complex Numbers

Addition and Subtraction

Let z=a+bi,  w=c+diz=a+bi,\ \ w=c+di:
  • z+w=(a+bi)+(c+di)=(a+c)+(b+d)iz+w=(a+bi)+(c+di) = (a+c) + (b+d)i
  • z−w=(a+bi)−(c+di)=(a−c)+(b−d)iz-w=(a+bi)-(c+di) = (a-c) + (b-d)i
In other words, we add/subtract like terms: keep the real parts together, and keep the imaginary parts together.
Example
Compute the sum of the complex numbers 3+2i3+2i and −1+5i-1+5i.
(3+2i)+(−1+5i)=3+(−1)+2i+5i=2+7i(\colorOne3+2i)+(\colorOne{-1}+5i) = \colorOne3+(\colorOne{-1}) + 2i+5i = 2 + 7i
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Multiplication

Expand using the usual rules of multiplication (keeping in mind i2=−1i^2=-1):

(a+bi)(c+di)=a(c+di)+bi(c+di)=ac+a(di)+(bi)c+(bi)(di)=(ac−bd)+(ad+bc)i\begin{aligned} (a+bi)\colorTwo{(c+di)} &= a\colorTwo{(c+di)} + bi\colorTwo{(c+di)}\\[0.5em] &= a\colorTwo c+a\colorTwo{(di)}+(bi)\colorTwo c+(bi) \colorTwo{(di) }\\[0.5em] &= (a\colorTwo c-b\colorTwo d)+(a\colorTwo d+b\colorTwo c)i \end{aligned}
Example
Compute (6−2i)(1+3i)(6-2i)(1+3i).

(6−2i)(1+3i)=6(1)+(6)(3i)+(−2i)(1)+(−2i)(3i)=6+18i−2i−6i2=6+16i−6(−1)=12+16i\begin{array}{rcl} (6-2i)(1+3i)&=& 6(1)+(6)(3i)+(-2i)(1)+(-2i)(3i)\\[0.5em] &=& 6+18i-2i-6i^2\\[0.5em] &=&6+16i-6(-1)\\[0.5em] &=&12+16i\\[1.5em] \end{array}
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Complex Conjugate

The complex conjugate of z=a+biz=a+bi is the complex number
zˉ=a−bi\boxed{\quad \bar{z}=a−bi \quad}
On the complex plane, zˉ\bar{z} is the reflection of zz across the real axis:

Properties
Let z,w∈Cz,w \in \Complex with z=a+biz = a+bi.
  • zzˉ=(a+bi)(a−bi)=a2+b2z\bar{z}=(a+bi)(a−bi)=a^2+b^2
  • (z±w)‾=z‾±w‾\overline{(z\pm w)}=\overline{z}\pm\overline{w}
  • (zˉ)‾=z\overline{(\bar{z})}=z
  • (zw)‾=zˉwˉ\overline{(zw)} = \bar z \bar w
  • (zw)‾=zˉwˉ\overline{\left(\dfrac{z}{w}\right)} =\dfrac{\bar{z}}{\bar{w}}
  • If Im(z)=0\text{Im}(z)=0, then z‾=a\overline{z}=a
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Division

To divide complex numbers, "rationalize" the denominator by multiplying top and bottom by the conjugate:
a+bic+di=a+bic+di⋅c−dic−di=(a+bi)(c−di)c2+d2\dfrac{a+bi}{c+di} = \dfrac{a+bi}{c+di}\cdot\dfrac{\colorTwo{c-di}}{\colorTwo{c-di}} = \dfrac{(a+bi)(c-di)}{c^2+d^2}
Example
Given z=1+i,  w=2+3iz=1+i,\ \ w = 2+3i, compute zw\dfrac{z}{w}.
1+i2+3i=1+i2+3i⋅2−3i2−3i=(1+i)(2−3i)(2)2+(3)2=2−3i+2i−3i213=(2−3(−1))+(−3+2)i13=5−i13\begin{aligned} \dfrac{1+i}{2+3i} &= \dfrac{1+i}{2+3i}\cdot\dfrac{\colorTwo{2-3i}}{\colorTwo{2-3i}}\\[1.5em] &=\dfrac{(1+i)(2-3i)}{(2)^2+(3)^2}\\[1.5em] &=\dfrac{2-3i+2i-3i^2}{13}\\[1.5em] &=\dfrac{\Big(2-3(-1)\Big)+\Big( -3 + 2 \Big)i}{13}\\[1.5em] &=\dfrac{5-i}{13}\\[1.5em] \end{aligned}
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Modulus

The modulus (or absolute value) of a complex number z=a+biz=a+bi is defined as:

∣z∣=a2+b2\boxed{\quad \lvert z\rvert =\sqrt{a^2 + b^2} \quad}
On the complex plane, the modulus is the length of the vector from 0⃗\vec 0 to zz:

Properties
Let z,w∈Cz,w \in \Complex with z=a+biz = a+bi.
  • ∣z∣=zzˉ(since zzˉ=a2+b2)\left|z\right|=\sqrt{z\bar{z}} \quad (\text{since } z\bar z = a^2 + b^2)
  • ∣zw∣=∣z∣∣w∣\left|zw\right|=\left|z\right|\left|w\right|
  • ∣z‾∣=∣z∣\left|\overline{z}\right|=\left|z\right|
  • If Im(z)=0\text{Im}(z)=0, then ∣z∣=∣a∣|z|=|a|
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Example: Basics of Complex Numbers

Perform the following operations.

Part A) (6−2i)+(i)(−4−5i)(6-2i)+(i)(-4-5i)

(6−2i)+(i)(−4−5i)=6−2i−4i−5i2=6−2i−4i+5=11−6i\begin{array}{rcl} (6-2i)+(i)(-4-5i) &=& 6-2i-4i-5i^2\\[0.5em] &=&6-2i-4i+5\\[0.5em] &=& 11-6i\\[0.5em] \end{array}

Part B) i1+i−1−ii\dfrac{i}{1+i}-\dfrac{1-i}{i}

i1+i−1−ii=i(i)−(1−i)(1+i)(1+i)(i)(cross multiply for common denominator)=i2−(12−i2)i+i2=−1−(1−(−1))i−1=−3−1+i⋅−1−i−1−i(rationalize denominator)=3+3i(−1)2−i2=3+3i2\begin{array}{rcl} \dfrac{i}{\colorTwo{1+i}}-\dfrac{1-i}{\colorOne i} &=& \dfrac{i(\colorOne i) - (1-i)(\colorTwo{1+i})}{(\colorTwo{1+i})(\colorOne i)} \quad\text{(cross multiply for common denominator)}\\[1.5em] &=& \dfrac{i^2 - (1^2-i^2)}{i+i^2}\\[1.5em] &=& \dfrac{-1 - (1-(-1))}{i-1}\\[1.5em] &=& \dfrac{-3}{-1+i} \cdot \colorFour{\dfrac{-1-i}{-1-i}} \quad\text{(rationalize denominator)}\\[1.5em] &=& \dfrac{3+3i}{(-1)^2-i^2}\\[1.5em] &=& \dfrac{3+3i}{2}\\[1.5em] \end{array}
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Part C) (2−i)(2+i)‾\overline{(2-i)(2+i)}

(2−i)(2+i)‾=(2)2+(1)2‾=(5)‾=5\begin{array}{rcl} \overline{(2-i)(2+i)}&=&\overline{(2)^2+(1)^2}\\[0.5em] &=& \overline{(5)}\\[0.5em] &=& 5 \end{array}
The conjugate of a real number is simply the real number itself (no imaginary part to negate).

Part D) ∣1−3i−1+3i∣\left|\dfrac{1-3i}{-1+3i}\right|

∣1−3i−1+3i∣=∣1−3i∣∣−1+3i∣=(1)2+(−3)2(−1)2+(3)2=1010=1\begin{array}{rcl} \left|\dfrac{1-3i}{-1+3i}\right| &=& \dfrac{|1-3i|}{|-1+3i|}\\[1.5em] &=& \dfrac{\sqrt{(1)^2+(-3)^2}}{\sqrt{(-1)^2+(3)^2}}\\[1.5em] &=& \dfrac{\sqrt{10}}{\sqrt{10}}\\[1.5em] &=& 1 \end{array}

Practice: Operations with Complex Numbers

Perform the following operations.
Simplify: 12i[(−1)(−i)+(1−i)]\dfrac{1}{2i}[(-1)(-i) + (1-i)]
Extra Practice
Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers

Basics of Complex Numbers
Practice: Operations with Complex Numbers
Perform the following operations.
Given the following complex number
z=3+4i1−iz = \frac{3 + 4i}{1 - i}
a) Write zz in Cartesian coordinates z=a+biz = a + bi .
Basics of Complex Numbers
Practice: Operations with Complex Numbers
Perform the following operations.
If z=3+2iz = 3+2i and w=1−4iw = 1- 4i, compute z−1z^{-1}and zwzw.
If z=3+2iz = 3+2i and w=1−4iw = 1- 4i, compute z−1z^{-1}and z⋅wz\cdot w
Simplify the expression 21+i+11+3i3−i\frac{2}{1+i}+\frac{11+3i}{3-i}
Practice: Operations With Complex Numbers
Perform the following operations:
(a) (−1)(−i)+(−i)(-1)(-i) + (-i)
(b) i(a+bi)‾\overline{i(a+bi)}
Complex Numbers
A complex number z∈Cz \in \mathbb{C} is a number of the form z=a+biz = a+bi where a and b are real numbers and i=p−1i = p−1. The number a is called the real part of zz and the number b is called the imaginary part of z.z.
Operations on Complex Numbers Consider two complex numbers z1=a1+ib1z1 = a_1 + ib_1 and z2=a2+ib2z_2 = a_2 + ib_2. Then
1. [Addition and Substraction] =z1±z2=(a1±a2)+i(b1±b2)=z_1 ± z_2 = (a_1 ± a_2) + i(b_1 ± b_2)