• The geometric representation of complex numbers: we represent the complex number a+bi as the point (a,b) in the coordinate plane (called “the complex plane” when we are talking about complex numbers). Better: we really think of a+bi as the vector which starts at the origin (0,0) and ends at (a,b).
• Review of operations on complex numbers.
• The absolute value (or modulus) of a complex number
Definition: |a+bi| is the distance between the point (a,b) and the origin (0,0) on the complex plane.
From the Pythagorean Theorem we find that we can compute
• The polar form of a complex number
The polar form has two coordinates:
, which is the angle in standard position to the vector ending at (a,b).
We can find by using the fact that , as you can see by looking at the graph.
• Changing from rectangular form (“standard form”) a+bi to polar form and vice-versa.
• Multiplying and dividing complex numbers using the polar form.
Note: I do not know of any application of complex numbers which uses the polar form with an angle in degrees. As far as I can determine, only radians are used in applications (to electrical engineering and various branches of physics). This is partly because the “angle” is usually time or some other one-dimensional quantity, not literally an angle. Also, in calculus it is much easier and more natural to work in radians. Therefore you should give all answers to the exercises in this section using radians for the angles.
For some information about the use of complex numbers in electrical engineering, and also a quick review of all of the above topics, you may want to view these documents:
Introduction to Complex Numbers in Physics/Engineering (I cannot figure out who the author of this very nice document is, but it comes from the Physics Department at College of Saint Benedict and Saint John’s University in Minnesota)
Complex Numbers and Phasors (by Professor Andrew E. Yagle, EECS 206 Instructor, Fall 2005 Dept. of EECS, The University of Michigan, Ann Arbor) [EECS = electrical engineering and computer science]
Note: The usual convention is that the angle (called the argument of the complex number) should be in the interval . However, this is not crucial. But make sure that your angle terminates in the correct quadrant!
• Review the definitions and examples worked in class. Take advantage of this opportunity to review finding exact values of trig functions and inverse trig functions using the basic triangles and the points on the axes from the unit circle!
• Do the assigned problems from Session 21
• Do the WeBWorK – due tomorrow by 11PM, so start early!
• No Warm-Up until next week
• There is a new review problem on WebWorK, also due tomorrow by 11 PM. Again, this is optional and will count as extra credit to your WeBWorK score if you do it.
• Don’t forget that Test 4 is scheduled for next Monday 12 May. The review problems were handed out in class and will also be available, along with the answers, on Piazza.