A complex number in the form

Z=x+iy

is said to be in rectangular, or Cartesian, form. Note that x and y are the coordinates of the point represented x+iy in the complex Cartesian plane.

Points in the complex plane can also be represented with polar coordinates. Recall that

define the relationship between polar and Cartesian coordinates. Substituting these into the rectangular form of a complex number, we get

for r≥0, 0≤θ< 2 π. θ is referred to as the argument of Z.

Examples

Express

in polar form.

Explanation

The modulus, or magnitude, of Z is defined as

It is the distance from the origin to the point (x,y). Recall that

This implies that

Product and Quotient of Complex Numbers in Polar Form

For complex numbers

and

If

then

Examples                                                         Explanation

Try these exercises

Solve. Round all decimals to the nearest hundredth

Answers to questions: