The Square Root of a Complex Number in Cartesian Form
Leads to: The general exact trigonometric forms for 45, 22.5, 11.125, etc.
So when you take a square root of a complex number, generally you would do it in cis form, where you can apply De Moivre's Theorem and let n=1/2. However, suppose you either are a masochist and don't wanna do that, or love exact values and your complex number doesn't have a nice cis form. THe algebra can be a bit messy, and takes a while to do, so it helps to have a formula to call on.
To start, we want to know what values of m and n satisfy this equation:
Or in other words:
This next part is a little messy. First, we are gonna solve for m:
Now with this, we can solve for n:
We now have both terms for n and m, defined in terms of a and b. Meaning, we now have an expression for
Written in terms of just a and b :D - Substituting in our values will give us:
It isn't the most neat looking thing in the world, but it certainly gets the job done!!
One final note about this, is that this formula is the cartesian equivalent of:
Which implies the following relations:
However, that is a relation to be explored on another day :)
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This is part two of a story I'm generating randomly! Read from the beginning here!
This entry is taking a twist from the Plot Twist Generator, which is:
"There is a sudden snowstorm, complicating the plans of the antagonist."
Which means I also needed an antagonist, so per the Villain Generator:
"This courageous wizard is motivated by vengeance. He employs forbidden magic in his plans, often unleashing unspeakable dark forces to achieve his goals. He wields an unusual artifact."
So I figured out what my next theme is. This month is NaNoWriMo, in which writers have to write a 50,000 word novel. Well, if a picture is worth 1000 words, that means I only need to tell a story in 50 pictures, right?
Throughout this month, I'm going to be telling a story solely through customization images, and short prompts generated by Seventh Sanctum. My starting prompt is from the Story Generator, and is:
"The story is about a courageous barbarian and a healthy cab driver. It takes place in a condominium in an outpost. The story begins with an eavesdropping and ends with someone going to sleep."
Every second post (posts will have 2-3 customization images per) will have a twist generated by the Plot Twist Generator, and I'm going to try and choose the first one that even slightly makes sense. Will this lead to a cohesive story? Probably not!
I've already generated my first few twists, so I can introduce my lead actors and their roles:
