2Complex Numbers Argand Diagram Recall that a complex number has a real component and an imaginary component.z = a + biImaginary axisaReal axisbiz = 3 – 2iz = 3 – 2iThe absolute value of a complex number is its distance from the origin.The names and letters are changing, but this sure looks familiar.
4How is it Different?In a rectangular system, you go left or right and up or down.In a trigonometric or polar system, you have a direction to travel and a distance to travel in that direction.
5Converting from Rectangular form to Trig form Convert z = 4 + 3i to trig form.1. Find r2. Find3. Fill in the blanks
6Converting from Trig Form to Rectangular Form This one’s easy.Evaluate the sin and cos.Distribute in rConvert 4(cos 30 + i sin 30) to rectangular form.1. Evaluate the sin and cos2. Distribute the 4.
7Multiplying Complex Numbers To multiply complex numbers in rectangular form, you would FOIL and convert i2 into –1.To multiply complex numbers in trig form, you simply multiply the rs and add the thetas.The formulas are scarier than it really is.
11De Moivre’s Theorem If is a complex number And n is a positive integer Then
12Who was De Moivre?A brilliant French mathematician who was persecuted in France because of his religious beliefs. De Moivre moved to England where he tutored mathematics privately and became friends with Sir Issac Newton.De Moivre made a breakthrough in the field of probability (writing the Doctrine of Chance), but more importantly he moved trigonometry into the field of analysis through complex numbers with De Moivre’s theorem.
13But, can we prove DeMoivre’s Theorem? Let’s look at some Powers of z.
19Powers of Complex Numbers This is horrible in rectangular form.It’s much nicer in trig form. You just raise the r to the power and multiply theta by the exponent.The best way to expand one of these is using Pascal’s triangle and binomial expansion.You’d need to use an i-chart to simplify.
20Roots of Complex Numbers There will be as many answers as the index of the root you are looking forSquare root = 2 answersCube root = 3 answers, etc.Answers will be spaced symmetrically around the circleYou divide a full circle by the number of answers to find out how far apart they are
21The formula k starts at 0 and goes up to n-1 Using DeMoivre’s Theorem we getk starts at 0 and goes up to n-1This is easier than it looks.
22General Process Problem must be in trig form Take the nth root of r. All answers have the same value for r.Divide theta by n to find the first angle.Divide a full circle by n to find out how much you add to theta to get to each subsequent answer.
23Example 1. Find the 4th root of 81 2. Divide theta by 4 to get the first angle.3. Divide a full circle (360) by 4 to find out how far apart the answers are.List the 4 answers.The only thing that changes is the angle.The number of answers equals the number of roots.