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.

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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.

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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.

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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.

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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.

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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.

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January 24, 2021 at 12:02am

#1002690

Comoedia Maxima

PROMPT January 24th

Write about your most memorable or unique teacher from the years you were in school. What made them so interesting and what do you remember about them the most?

Well, I could talk about the science teacher I had in 8th grade who insisted -- and threatened me with the principal's office if I continued to contradict her -- that the reason Earth has gravity is because it rotates.

Look, I get it, not everyone can know everything. And if I'd heard that from an English or social studies teacher, I might have been able to write it off. But a science teacher? No wonder we're a nation of dumbshits.

But apart from that really quite abysmal gap in her knowledge, she was otherwise fairly forgettable.

So I'm going to go with my high school Latin teacher, Ms. P.

She wasn't my first Latin teacher. Freshman year, it was an older lady who'd been around the arena a time or two and brooked no shit from her students.

Second year, et cetera? New teacher. And by "new," I mean fresh out of ed-school.

I don't remember much Latin, but I do remember the scent of blood in the water.

Now, don't get me wrong; we all liked Ms. P. She was close enough to our ages to be somewhat relatable, but not so attractive as to be distracting to us boys. Gods, if we hadn't actually liked her, I can't even imagine the hell we would have put her through. As it was, it was mostly just harmless teenage pranks and antics.

Mostly.

Ms. P also taught math, but it was remedial math so, not to brag or anything, I wasn't in those classes. I mention this only because the math and science departments were clear on the other side of the school from the language classrooms. And, I guess, with her being a new teacher, the evil genius who did the scheduling put her math period immediately before my Latin class that first year she taught.

This gave us plenty of time to hone our comedy skills before she showed up.

At the end of a day, she'd put an English vocabulary word on the blackboard for us to ponder prior to her arrival the following day. (Why English? Well, in case you haven't noticed, over half of the words in English have Latin roots; the only reason it's considered a Germanic language is the sentence structure. So the vocabulary word was, I suppose, an effort to provide a reason why learning Latin is relevant.) So we did ponder the word - usually by writing a sentence wherein that word was used as a pun, and/or by providing a punny definition. Example: "Bacteria." "The rear part of a cafeteria."

Yes, I know "bacteria" was from ancient Greek, not ancient Latin, but it's the only one I could remember after all these years. Don't be pedantic.

Then there was Rufus Roman.

Rufus Roman was a stick figure with one of those brush-top helmets you see in movies about ancient Rome, and he held a gladius and a scutum.(sword and shield). We always drew him, on the blackboard, with either a big smile on his face, or, sometimes, with an expression of abject terror (when facing his archenemy, Barney Barbarian.) (Barney, of course, wore a helmet with horns, and an axe, and sported a beard and a mean, toothy, growly face.)

This was, and still is, the entire range of my so-called artistic talent. It never developed further. Fortunately, my comedic talents did. Well. Sort of. Maybe. You can decide that for yourself, and keep it to yourself.

Anyway, we could all tell that Ms. P. was trying very, very hard to maintain discipline in class by not acknowledging the comedy gold mine she'd walk into every time she'd march over from the other side of the school and look at the board. But she didn't succeed. Sometimes she even laughed out loud before she composed herself and very pointedly erased our masterworks.

One day, I think maybe it was because she'd forgotten to post the vocabulary word, we spent the five minutes we had before the teacher walked in by turning every piece of furniture in the room 180 degrees. As I recall, this involved a couple of filing cabinets, all of the student desk/chairs, the teacher's desk and wheeled chair, a lectern, and probably a few other odds and ends. There were like 8 of us in the class so it didn't take very long. And when she came in, at first she didn't notice.

Now, it's important to note two things: One, every day, she'd come in, sit down, stretch her arms to the sides of her desk, and wheel herself forward so her legs were under the desk. And two, the desk itself had a barrier on the student side. Well, so after everything got turned around, she simply swiveled the chair around, sat down, grabbed the desk, pulled herself forward and BANG her knees hit the wall on the front of the desk.

Fortunately for us, Ms. P. had a decent sense of humor and a high tolerance for pain -- both necessary qualities for a high school teacher to have.

She'd be in her 60s now, I guess. If she's still teaching, I bet she no longer brooks any shit whatsoever from her students.

We taught her well.

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