# 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 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\$

### 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 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\$.

### 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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