Converting Complex Numbers
Rectangular to Polar
|Here are the formal expression for converting rectangular to polar coordinates: |
For the sake of this lesson, lets assume that the two forms of complex numbers are expressed this way:
- Rectangular: a +jb
- Polar: cÐq
The objective is to express a +jb in in the form of cÐq.
Recall how the same complex number can be represented two different ways:
Citing a complex number with rectangular coordinates.
Citing the same complex number with polar coordinates.
Overlay the two coordinate system to show that there is a right triangle made up of all the important parts of the complex number. Converting from rectangular to polar coordinates is a matter of solving the triangle for the hypotenuse (c) and angle q.
a = the length of the base of the triangle
b = the height of the triangle
c = the hypotenuse of the triangle
q = the angle
From the Pythagorean theorem:
That gives us the length component of the polar form in terms of the rectangular components, a and b.
From basic trigonometry:
That provides the angle component of the polar form, also in terms of the rectangular components.
Converting from rectangular to polar coordinates for any complex number is a matter of solving those two equations.
Convert 13.5 +j7.5 to polar form
Step 1. Identify the components
- a = 13.5
- b = 7.5
Step 2. Solve for the polar length
Step 3. Solve for the polar angle:
Step 4. Put it all together
13.5 +j7.5 = 15.4Ð29°
Learn From More Examples
- 10 +j10 = 14.1Ð45°
- 10 -j10 = 14.1Ð-45°
- 1 + j3 = 3.16Ð71.6°
- 2 + j20 = 20.1Ð84.3
- 20 +j2 = 20.1Ð5.7°
- 10 = 10Ð0°
- j10 = 10Ð90°
- 3 +j5 = 5.8Ð59°
- 5 +j2 = 5.4Ð21.8°
- 3 -j10 = 10.4Ð-73.3°