Finding Magnitude of complex number expression

[mkematt96]

Homework Statement

We are given Z, and are asked to find the magnitude of the expression. See attached picture(s)

Homework Equations

See attached pictures(s)

The Attempt at a Solution

When I solved it on the exam, I did it the long way using De Moivre's theorem. I ended up making a few sign errors which cost me points. When my professor went over the exam, he did the problem as shown on the second picture with the purple pen writing. What I am wondering is why you can solve it this way? Why in the denominator you can just multiple the magnitude of both terms without having to evaluate it?

 

[Mark44]

Homework Statement

We are given Z, and are asked to find the magnitude of the expression. See attached picture(s)

Homework Equations

See attached pictures(s)

The Attempt at a Solution

When I solved it on the exam, I did it the long way using De Moivre's theorem. I ended up making a few sign errors which cost me points. When my professor went over the exam, he did the problem as shown on the second picture with the purple pen writing. What I am wondering is why you can solve it this way? Why in the denominator you can just multiple the magnitude of both terms without having to evaluate it? View attachment 203453 View attachment 203454

Because, for example, $|\frac z 2 | = \frac {|z|} 2$. The denominator in the original problem is a real number. The work done in just a few lines (in purple) is extension of my example. Being much simpler, it's the better approach.

 

[scottdave]

More generally, $|\frac z w | = \frac {|z|} {|w|}$. For example, if z = |z|e^i*argz and w = |w|e^i*argw, then you can write z / w as (|z|/|w|)*e^{i*(argz-argw)}.Since e^i*a has a magnitude of 1, then it has no effect on the magnitude.

In general when you multiply two complex numbers, you multiply the magnitudes and add the angles. Dividing, you divide the magnitudes and subtract the angles.