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

October 4, 2022 at 12:02am

#1038578

Loyalty Ad

Even as a kid, I had issues with this.

How the Pledge of Allegiance Went From PR Gimmick to Patriotic Vow
Francis Bellamy had no idea how famous, and controversial, his quick ditty would become

My biggest issue was expecting second-graders to know what words like "allegiance" meant.

On the morning of October 21, 1892, children at schools across the country rose to their feet, faced a newly installed American flag and, for the first time, recited 23 words written by a man that few people today can name. “I pledge allegiance to my Flag and to the Republic for which it stands—one nation indivisible—with liberty and justice for all.”

While this article (actually an ad for a book but it still contains interesting information) is from 2015, I note that this month marks the 130th anniversary of that day. That's a bit more than half of the country's existence.

Francis Bellamy reportedly wrote the Pledge of Allegiance in two hours...

What did he do, laboriously engrave it into granite? Hopefully it was American granite.

In a marketing gimmick, the Companion offered U.S. flags to readers who sold subscriptions, and now, with the looming 400th anniversary of Christopher Columbus’ arrival in the New World, the magazine planned to raise the Stars and Stripes “over every Public School from the Atlantic to the Pacific” and salute it with an oath.

"A marketing gimmick."

I mean, really, can you think of anything more quintessentially American than that? Okay, anything that doesn't involve firearms?

Bellamy, a former Baptist preacher, had irritated his Boston Brahmin flock with his socialist ideas.

I just want to leave this part here for contemplation.

In a series of speeches and editorials that were equal parts marketing, political theory and racism, he argued that Gilded Age capitalism, along with “every alien immigrant of inferior race,” eroded traditional values, and that pledging allegiance would ensure “that the distinctive principles of true Americanism will not perish as long as free, public education endures.”

Okay, maybe rampant capitalism, anti-immigrant sentiment, racism, and jingoism are right up there with marketing gimmicks as true American values. We're trying to destroy free public education, though, so there's that.

In 1954, as the cold war intensified, Congress added the words “under God” to distinguish the United States from “godless Communism.”

Because clearly that's the only difference.

It's rare that I'm in a position to endure the Pledge these days, but when I do, I omit that part on general principle.

That it was a preacher who wrote the thing, and didn't include any references to any deity in it, should be cause for thought. But again, this is America; thought is for commies and pussies.

The snappy oath first printed in a 5-cent children’s magazine is better known than any venerable text committed to parchment in Philadelphia.

Oh, I don't know. The Schoolhouse Rock rendition of the Preamble to the Constitution is an earworm that's firmly etched in my memory.

Yet the pledge continues to have its critics, with some pointing out the irony of requiring citizens to swear fealty to a nation that prizes freedom of thought and speech.

On paper.

The historian Richard J. Ellis, author of the 2005 book To the Flag: The Unlikely History of the Pledge of Allegiance, acknowledges that the oath is “paradoxical and puzzling,” but he also admires the aspirational quality of its spare poetry. “The appeal of Bellamy’s pledge is the statement of universal principles,” he says, “which transcends the particular biases or agendas of the people who created it.”

What universal principles? That a flag exists? "Republic," "indivisible," "liberty and justice" are all abstract concepts and hardly universal.

So is "nation," for that matter.

Bellamy did some transcending of his own. The onetime committed socialist went on to enjoy a lucrative career as a New York City advertising man, penning odes to Westinghouse and Allied Chemical and a book called Effective Magazine Advertising.

Ah, yes, the sure cure for socialism: money.

But his favorite bit of copy remained the pledge—“this little formula,” he wrote in 1923, with an ad man’s faith in sloganeering, which “has been pounding away on the impressionable minds of children for a generation."

Sure, he didn't invent indoctrination. But he nationalized it.

Despite my issues with making impressionable kids recite a loyalty oath, though, I can't fault the ideals of "liberty and justice for all." But I think we have to acknowledge that those ideals are something to work towards, not reflective of the current state of affairs. Especially knowing that the guy who wrote the words wasn't interested in the "for all" part, going by his statements on "inferior races." (Just to be clear, no, I'm not trying to "cancel" him. Sucks that I have to issue this disclaimer, but these days that's what some people will infer.)

I vaguely remember saying a while back that the Coca-Cola Santa Claus was the most successful marketing campaign of all time. I'm big enough to admit that I was wrong: the Pledge holds that title.

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