Science Library - free educational site

Complex numbers as vectors

Complex Numbers as 2-D Vectors

A complex number contains a real part, $a = Re(z)$, and an imaginary part, $b=Im(z)$. $a$ and $b$ are both real numbers, and $z=a + bi$ is the Cartesian form, since it allows the complex number to be plotted on a 2D Cartesian-like coordinate system.

Argand
Argand diagram for complex numbers represented as 2D vectors

Argand Diagram

Named after Argand, an Argand plane, or diagram, plots the real part of a complex number along the x-axis, and the imaginary part along the y-axis.

  • ${OP}↖{→} = [\table 4;2]$ represents $4 + 2i$
  • ${OQ}↖{→} = [\table -1/2;5/2]$ represents $-1/2 + 5/2i$
Argand Plane question

Example of vector addition in the Argand Plane

a) $z_1 + z_2 = (2+i) + (-3+3i) = -1 + 4i$

b) $z_1 - z_2 = (2+i) - (-3+3i) = 5 - 2i$

Modulus, Argument, Polar Form

Modulus

Modulus of a vector
Modulus of a complex number vector

The modulus of a complex number $|z|$ is the magnitude of the vector $[\table a;b]$, where $z=a+bi$.

The modulus is therefore found from Pythagoras: $|z| = √{a^2+b^2}$.

Modulus rules

$|z^{*}|=|z|$

$|z|^2=zz^{*}$

$|z_1z_2|=|z_1||z_2|$

$|{z_1}/{z_2}| = {|z_1|}/{|z_2|}$, if $z_2≠0$

$|z_1z_2{z_3}...z_n| = |z_1||z_2||z_3|...|z_n|$

$|z^n|=|z|^{n}$ for $n$ a positive integer

Distance in the Number Plane

$|z_1-z_2|$ is the distance between points $P_1$ and $P_2$, where $z_1 ≡ {OP_1}↖{→}$ and $z_2 ≡ {OP_2}↖{→}$.

The midpoint of the resulting vector can be found by dividing by 2: mid-point between $P$ and $Q$ is: ${OP↖{→} + OQ↖{→}}/2$.

Argument

The angle θ the complex number vector z makes with the real (x) axis is the Argument (arg z).

Since when $θ=0$, the length of the vector has infinite possibilities, z is not a function, unless we specify that $θ ∈]-π, π]$, which is one full revolution.

Real numbers have argument of $0$ or $π$.

Purely imaginary numbers have argument of $π/2$ or $-π/2$.

Polar Form

Euler's Polar Coordinates

$$z=re^{iθ}$$

where $r=|z|$ and $θ$ = arg$z$.

The polar coordinates of a point in space are an alternative to the Cartesian coordinate system ($z=x+yi$), and expresses part of the location as the angle to the positive x-axis.

Given a vector $z$, the real part is $r⋅cosθ$, where $r$ is the length and θ is the angle to the x-axis (argument).

Similarly the imaginary part is $r⋅sinθ$.

$z = r⋅cosθ + r⋅isinθ$.

Since $r$ is the modulus of the vector, $|z|$, we can express the polar coordinates of a vector as:

$$z = |z|cisθ$$

where $cisθ$ is a shorthand way of writing $cosθ + isinθ$.

The polar form is useful for finding the powers and roots of numbers, and other operations, such as multiplication and division.

Example

The vector $z=1+i$ forms a right-angle triangle in the Argand plane. Its modulus is the hypotenuse, i.e. $|z| = √2$, and the argument (angle) is 45°, or $θ = π/4$.

Cartesian form: $z=1+i$

Polar form: $z=√2cisθ$

Multiplication and Division in Polar Form

$cisθ×cisΦ = cis(θ+Φ)$

${cisθ}/{cisΦ} = cis(θ-Φ)$

$cis(θ+k2π) = cisθ$

If $z_1 = r_1$cis$θ_1$ and $z_2 = r_2$cis$θ_2$, then:

$$z_1z_2 = r_1r_2cis(θ_1 + θ_2)$$

Expressed in Euler form:

$$(r_1e^{θ_{1}i})(r_2e^{θ_{2}i}) = r_1r_2e^{(θ_{1}+θ_{2})i}$$

If $z = r$cis$θ$, then $1/z = 1/r$cis$(-θ)$, $r≠ 0$.

$${z_1}/{z_2} = (r_1cisθ_1 )(1/{r_2}cis(-θ_2) ) = {r_1}/{r_2} cis(θ_1-θ_2), r≠ 0$$

Since $1/{re^{θi}} = 1/{r}e^{-θi}$, then:

$${r_1e^{θ_1i}}/{r_2e^{θ_2i}} = {r_1}/{r_2} e^{(θ_1-θ_2)i}$$

De Moivre's Theorem

De Moivre's theorem allows a complex number to be expressed as integral powers:

$$[r(cosθ + i⋅sinθ)]^n = r^n(cos(nθ) + i⋅sin(nθ))$$

where r ∈ $ℝ^{+}$, θ ∈ ℝ and n ∈ ℤ

Content © Andrew Bone. All rights reserved. Created : October 12, 2014

Latest Item on Science Library:

The most recent article is:

Trigonometry

View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

Subject of the Week

Resources

Science resources on ScienceLibrary.info. Games, puzzles, enigmas, internet resources, science fiction and fact, the weird and the wonderful things about the natural world. Have fun while learning Science with ScienceLibrary.info.

Science

Great Scientists

Isaac Newton

1642 - 1727

Issac Newton is possibly the most influential scientist of all time. In the second half of the 17th century, he produced a breathtaking number of physics and mathematical laws and methods, explaining forces and physical phenomena, and deriving mathematical explanations still in use today.

Isaac Newton, 1642 - 1727
ScienceLibrary.info

Quote of the day...

ZumGuy Internet Promotions

Yoga in Mendrisio