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🎶

about 22 hours 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

July 2, 2024 at 9:00am

#1073461

Imma Let You Finish, But...

The current round of "Journalistic Intentions"

[18+] features prompts relating to the Olympics. This is a subject I know almost nothing about, so I get to stretch my superpower of being able to write about any topic.

To make it even more potentially difficult for myself, I'll be picking the prompts (there will be eight of them, including this one) at random from the list.

Vanderlei de Lima

I've long thought of the Olympics as the exclusive province of young people. Well, in the sports, anyway; I imagine the IOC as a bunch of septuagenarians proclaiming this and that from on high (I don't know who actually composes that body).

And by "young," I generally mean "teens and early twenties." Then you retire, do commercials, and eventually get involved with a family who's famous for being famous. Hell if I know; it's so hard to Keep Up these days.

But, according to this guy's Wiki page,

dude was 35 when he ran in the 2004 games.

For context, I'm only three years older than him. For more context, Springsteen was 35 when he released Born in the USA (but much younger when he released Born to Run, which might have been more appropriate for Lima).

From that Wiki page:

Vanderlei Cordeiro de Lima (born 4 July 1969) is a Brazilian retired long-distance runner.

Now, see, I'd have guessed Peruvian from the name. Shows how much I know. (And no, I've never even heard his name before, to the best of my recollection.)

While leading the marathon after 35 km at the 2004 Summer Olympics, he was attacked on the course by Irish former priest Cornelius "Neil" Horan.

There's gotta be a story there. Fortunately, the article later expands upon it:

Soon after the 35 km (22 mi) mark, holding a lead of around 25 seconds, Lima was halted and grappled with by spectator Neil Horan, an Irish priest who was later defrocked.

The implication is that Horan was actively serving as a priest at the time, so saying he was attacked by a "former priest" up there is a bit misleading.

What Godly or worldly reason could Horan have had for being a human stumbling-block?

Horan said "I wasn’t doing it as a prank, I was doing it to spread the gospel and to prepare people for the second coming.”

Religious "logic," folks. I know if I wanted to convince someone my invisible sky-wizard was about to press the big red reset button, I couldn't think of a more appropriate way to spread the message than to fuck up someone's marathon run.

2004 was before social media infected the internet, but the internet itself was absolutely a thing then. I know this because that's the year I joined WDC, not long after the Olympics.

And here we are, 20 years later, and despite every effort and fervent prayers, the second ain't come yet.

In any case, there's not much else at the Wiki link besides Lima's other accomplishments in putting one foot in front of the other faster than everybody else. He did win sportsmanship awards, though, so I figure that counts for something regarding his character.

Which is more than I can say for his attacker.

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