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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

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