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How is a complex number represented using its modulus and argument?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

A complex number $z = a + b i$ can be represented using its modulus ( $r$ ) and argument ( $ϕ$ ) through the polar form. The modulus $r$ represents the absolute value or the distance of the number from the origin in the Argand diagram, while the argument $ϕ$ is the angle the vector makes with the positive real axis.

The relationship is expressed as:
$z = r (\cos ϕ + i \sin ϕ)$

This representation is deeply connected to Euler’s formula, which states that $e^{i ϕ} = \cos ϕ + i \sin ϕ$ . Consequently, a complex number can also be written in exponential form as:
$z = r e^{i ϕ}$

This approach allows for the efficient manipulation of complex numbers, particularly when calculating powers or roots, as demonstrated by Abraham de Moivre and Leonhard Euler in the development of modern analytic trigonometry.12