Argand Diagram, Modulus, and Polar Coordinates

The Argand diagram plots a complex number

$z = a+bi$

as the point

$(a, b)$

— the real part on the horizontal axis, the imaginary part on the vertical axis.

Two things follow naturally from that picture:

• The modulus

  $|z|$

  is the distance from the origin:

  $|z| = \sqrt{a^2+b^2}$

  .
• The argument

  $\theta$

  is the angle the point makes with the positive real axis, giving the polar form

  $z = r(\cos\theta + i\sin\theta)$

  where

  $r = |z|$

  .

Example:

$z = 1+i$

sits at

$(1,1)$

, so

$|z| = \sqrt{2}$

and

$\theta = 45^\circ$

, giving polar form

$\sqrt{2}(\cos 45^\circ + i\sin 45^\circ)$

.