Polar coordinatesThe representation of a complex number as a sum
of a real and imaginary number, z = x + iy, is called its
Cartesian representation. Recall from trigonometry that if x, y,
r are real numbers and r 2 = x 2 + y 2, then there is a unique
number θ with 0 ≤ θ < 2π such that sin(θ) = y / r. That number
is |θ||=||cos -1(x / r),| |=||sin -1(y / r)| |=||tan -1(y / x)|
We can therefore express any complex number z = x + iy as
|z||=||| z | (x / | z | + iy / | z |)| |=||| z | (cos θ + i sin
θ)| |=||| z | e iθ ,| where θ = tan -1(y / x). The angle or
argument θ is measured in radians, and it is written as arg(z).
So we have the polar representation of any complex number z as
The two representations are related by The values x and y are
called the Cartesian coordinates of z, while r and θ are its
polar coordinates. Note that r is real and r 3 0. Note that for
any integer K, This is because Thus, the polar coordinates (r,
θ) and (r, θ + 2Kπ) for any integer K represent the same complex
number. Thus, the pola