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.

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 $-1/2 + 5/2i$

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

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

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

$|z_1-z_2|$ 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: ${OP↖{→} + OQ↖{→}}/2$.

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 $-π/2$.

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.

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: $z=√2cisθ$

$cisθ×cisΦ = cis(θ+Φ)$

${cisθ}/{cisΦ} = cis(θ-Φ)$

$cis(θ+k2π) = cisθ$

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 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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b. 1939

Gro Harlem Brundtland, born 1939, is one of the most important figureheads and leaders of the environmental movement. In 1987, she chaired the famous Brundtland Commission for the UNO, which established Sustainable Development as a central goal of development and environmental protection.

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