# Math 0250 Homework 1

• 0.1 Number 4
a
b
c
d
e
• 0.2 Number 1
a
The point 3-4i is in the fourth quadrant, so you can use a polar angle between and , given by . It's distance from the origin is . Therefore

3-4i = 5 e-0.92730 i

b
The point -5+12i is in the second quadrant, so you can use a polar angle between and , given by . It's distance from the origin is . Therefore

-5+12i = 13 e1.9656 i

c
The point 3+4i is in the fourth quadrant, so you can use a polar angle between and , given by . It's distance from the origin is . Therefore

3-4i = 5 e0.92730 i

• 0.2 Number 4
a
b
c
• 0.2 Number 5
a
Write

b
Write

Taking k = 0, 1, 2 gives the three values

• C1 The computer gives for because .
• C2 The value of b is b = e10i = -0.83907 - 0.54402i and its log is . The reason you didn't get 10i for the log is that the computer represents e10i with a polar angle between and before taking the log, and so always returns a log with imaginary part between and . The number 10i is one of infinitely many value of . The value the computer gives is .
• C3 You should have gotten a bunch of points arranged around a circle of radius one and centered at the origin. That's because always sits on this circle.
• C4 You should have gotten a long list of 1's. That's because you're taking the absolute values of a bunch of points, each of which is exactly one unit away from the origin, and absolute value measures the distance from the origin.
• C5 You should have gotten a bunch of points arranged on a vertical line, to the right of the imaginary axis by units. That's because you're taking the log of a bunch of numbers of the form 2 eit, which gives a bunch of numbers of the form , each with real part .
• C6 Here's the convincing argument''. Write

with r=|A| and . Then

Taking real parts gives the formula. Are you convinced?
• C7 Just calculate. With A = a-bi,

which has real part as required.