Polar Form of Complex Numbers

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:

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