##\displaystyle arg(z) = tan^{-1}\left(\frac{Im(z)}{Re(z)}\right) \ ##Banker said:Homework Statement
Question 3.b. - http://imgur.com/ztLiRvx
Homework Equations
For the sake of simplicity, let's assume that lambda = x.
The Attempt at a Solution
I tried equating the real an imaginary parts of arctan(1/4).
Real: x/2 + 3 = 4. This gives x = 2.
Imaginary: x/2 - 3 = 1. This gives x = 8.
I'm getting two different values of x(lambda).
?Banker said:Homework Statement
Question 3.b. - http://imgur.com/ztLiRvx
Homework Equations
For the sake of simplicity, let's assume that lambda = x.
The Attempt at a Solution
I tried equating the real an imaginary parts of arctan(1/4).
No.Banker said:Real: x/2 + 3 = 4. This gives x = 2.
Imaginary: x/2 - 3 = 1. This gives x = 8.
Banker said:I'm getting two different values of x(lambda).
Yes, those are correct, although we generally like to see details of how you worked it out. That makes it easier for us to check your answer.Banker said:@Mark44 In my original attempt, I tried to reverse the process you mentioned, but as Sammy pointed out, it only tells the ratio.
So I then divided the equations for the imaginary and real parts in part 'a' and got 10 as my answer. So that will give 68 as the answer to the second part of 3.b, right?
Looks good.Banker said:
Complex numbers are numbers with both a real and imaginary component. The real component is a normal number, while the imaginary component is a multiple of the imaginary unit, i. Real numbers can be plotted on a number line, while complex numbers are plotted on a complex plane with the real component on the x-axis and the imaginary component on the y-axis.
To add or subtract complex numbers, simply combine the real components and combine the imaginary components separately. For example, (3 + 2i) + (5 - 4i) = (3 + 5) + (2i - 4i) = 8 - 2i. Similarly, (3 + 2i) - (5 - 4i) = (3 - 5) + (2i + 4i) = -2 + 6i.
Yes, you can multiply and divide complex numbers using the FOIL method (First, Outer, Inner, Last). For example, (3 + 2i)(5 - 4i) = 15 - 12i + 10i - 8i^2 = 15 - 2i - 8(-1) = 23 - 2i. To divide, you can multiply by the complex conjugate of the denominator to eliminate the imaginary component in the denominator.
The complex conjugate of a complex number is the same number with the sign of the imaginary component changed. For example, the complex conjugate of 3 + 2i is 3 - 2i. This is important in dividing complex numbers, as it allows us to eliminate the imaginary component in the denominator.
Complex numbers are used in many fields, such as engineering, physics, and economics. They are particularly useful in describing and analyzing oscillatory systems, such as electrical circuits and sound waves. They are also used in signal processing to analyze and filter signals. In economics, complex numbers are used in the study of complex systems, such as financial markets and population dynamics.