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[Mathematics] 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.

[Mathematics] 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: $z=√2cisθ$