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 24, 2021 at 12:03am

#1022227

I Did Nazi That

I want to say I respect all of my former teachers.

PROMPT November 24^th

In a previous prompt, I asked you to write about your
best, or favorite teacher. Tonight write about your darkest teacher.

I want to say I respect all my former teachers, but I can't.

I know I've spoken before about a 7th or 8th grade science teacher who was absolutely certain that it is rotation that causes gravity; that if the Earth weren't spinning, nothing would stay stuck to it. (This is a different issue from if the Earth were to suddenly stop spinning, which would be Bad for other reasons).

But okay, everyone is wrong about something, and most of us refuse to consider corrections. In this particular case, I could have forgiven it if it had been, say, an English teacher. But science? No.

Just to be clear, it's mass that produces gravity, not spin. I'm pretty sure she got confused from reading science fiction stories that featured spinning spaceships or whatever, which generate something akin to gravity on the inside as a result of their rotaion.

Still, apart from that blind spot, I don't think she was a terrible teacher.

In the same school, though, I once had a Nazi for a teacher for a whole year.

Now, keep in mind that this was a very long time ago (though not quite so long ago as WWII), so I don't remember a lot of details. It's mostly just the impression that I'm left with at this much later date. She was pretty old then, so I doubt she's still around, but just in case, I'm going to call her Fraulein Braun instead of whatever her real name was.

Most years, through elementary and middle school, teachers would be eager for the class to learn about something other than mainstream US Protestantism (the majority representation in a public school in rural Virginia), so anyone from a different culture was always encouraged to enlighten the class about their traditions. Like, they'd make me go up to the board and talk about Hanukkah every damn December. I got pretty good at drawing menorahs and dreidels with chalk. Anyway, other people talked about their experiences too. One guy was a Native American. Occasionally we'd have exchange students and such, from other countries.

Not F. Braun, though. Such things were suppressed. The other thing I remember clearly is going to the principal's office far more often than usual that year. Which was remarkable, as the principal already had a special chair with my name on it; it just got a lot more use that year.

She wasn't too overt about it, though. Couldn't be, I suppose, or she'd have been fired. It was a more subtle attitude, though it was obvious that the white kids got more of her help. I don't know if she fudged grades or anything; I didn't care as long as I didn't fail.

I also can't recall a thing about what we learned that year. I think I was too busy trying to keep the teacher from remembering I existed.

It's not like I hadn't already figured out that bigotry existed, by that time, so F. Braun didn't destroy my innocence or anything of that sort. It's just unfortunate that such people had positions of authority over a diverse group of students.

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