Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 17, 2021 at 12:01am

#1021786

The Butler Did It

Yeah, I'm going with the cliché here. Or am I?

PROMPT November 17^th

You have been invited to a posh dinner party hosted by one of the wealthiest people in your small town. When you arrive, your host is does not greet you, and the butler informs you they have been missing last night. A quick search by all the party goers finds him/her dead. Their body is in a small clearing in the woods behind the mansion. Being a modern day Nancy Drew/Joe Hardy type of person, you begin your own investigation. What do you find that leads you to the murderer. Who is the murderer? How was the host killed?

I mean, come on, everyone knows the butler did it. But since everyone knows that, obviously the butler wouldn't have done it because he'd be the prime suspect. But knowing that everyone knows the butler did it, thus removing suspicion from the butler, brings suspicion back on the butler because they know that he knows that they know that he knows...

Obviously, this is fiction, because I don't get invited to those sorts of parties. But see, the host was trying to make nice to me because he wanted to hire me as a ghost writer. What Mr. Grabcash didn't remember was that, about ten years ago, he hired me to do some web design work, which I did, and it was remarkable (for the time).

And the cheap bastard stiffed me out of my fee.

Since then, I've been waiting for a chance to exact revenge, but Grabcash is surrounded by flunkies, minions, and bodyguards, so what could I do?

Well, I took a page out of the disinformation playbook, and started spreading rumors. This was only partially successful, at first, as Grabcash became convinced that the gardener, Elrod, was boning Mrs. Grabcash. I was hoping to hear about a divorce, but no, all he did was fire Elrod, which wasn't my intent (I made sure to hook him up with a venture capitalist in another town, but he really did shag that one's wife and I have no idea what happened to him after that).

Thing is, though, I believe in asymmetric warfare and nuclear revenge -- provided, of course, that I'm on the giving end and not the receiving end. Still salty about the $799.95 he owed me -- closer to $1000 now, because of interest and late fees -- I had to hatch the perfect plan.

Grabcash liked to take a stroll around the grounds most evenings. I know in stories you always hear about "oh, so-and-so goes out at the same time every single day and does the exact same thing, so that's when we move." This is often the most unrealistic part of heist or mystery dramas, because seriously, no one is that regular, even if they eat nothing but Fiber One. Mr. G's evening constitutional schedule was nowhere near predictable, except that it was always sometime after 3 pm and before whatever time sunset happened to be at that time of year. If, of course, he bothered to go out at all.

Oh, yeah, how did I know this? Because I was boning Mrs. Grabcash, and she told me.

Anyway, the whole thing was her idea, which was a good thing, because she was the only one able to stash the silver and some of her jewelry in a secret compartment in the butler's quarters that even the butler didn't know about. After what happened to Elrod, clearly Mr. G would, upon finding out about the butler's secret stash (which of course even the butler wasn't aware of), fire the butler, and of course the butler (I suppose I should note that his name was Aragorn, because even his parents hated him) didn't want to be fired, so he did the only thing that made any sense, which was to wait in hiding with Torgo the bodyguard's sniper rifle from Afghanistan. One shot to the head, bam, no more threat of being fired and Torgo (not his real name; he got it from a movie) gets fingered.

Only problem is Torgo didn't do it; that's an obvious setup. Once Aragorn's secret stash is discovered - much to everyone's surprise, especially Aragorn's, though everyone will think he's faking it - it will be absolutely clear that the butler did it, using Torgo's rifle which happened to have Aragorn's fingerprints all over the barrel because Mrs. G had put them there late one night while Aragorn slept.

No one knew I had sniper training; that was a different life and I've changed names three times since then. And no one could possibly suspect that anyone would off Mr. Grabcash over a measly $1000 unpaid bill, but hey, it's the principle of the thing. Plus now me and the former Mrs. G can scramble off to Tahiti once the dust settles.

I'll even give Torgo a raise.

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