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

September 23, 2021 at 12:01am

#1017903

Friend

No matter how much you learn about history, there is always more. This Cracked article, for example.

'The Public Universal Friend,' Or The 1776 US History Story You Never Learned In School

As per usual, the story's broken down into bite-sized, numbered chunks for those of us with limited atten-SQUIRREL!

6. Our Story Begins With Our Hero's Death

When you were in school, maybe they taught you something about Pilgrims coming to Massachusetts in 1620 because they, unlike those evil people back in England, believed in religious freedom. It's a nice story that falls apart a little when you hear the settlers soon turned to expelling and even killing people for their religion.

Now they're stealing my schtick. I've been saying things like this for years.

Many Quakers moved to Rhode Island, where they responded to their past persecution by setting up their own restrictive society, as is tradition.

They're getting to the point, I promise.

Amy and Jeremiah Wilkinson had 12 kids. They named child number eight Jemima, after a daughter God gave Job after killing his other kids to win a bet, which is the sort of thing that might set a kid up for a religious life. In October 1776, while founders were passing around and signing a declaration of independence, 23-year-old Jemima caught a serious disease, possibly typhus. This disease proved fatal.

No, no relation.

Two archangels greeted the soul in heaven then announced that God had a new plan, as well as a new identity to confer unto the departed. And so Jemima awoke and answered to "Jemima" no more. Now, they were to be known as "The Public Universal Friend."

Remember this story next time you're tempted to think of gender dysphoria as a 21st century phenomenon.

5. A Ministry With Absolutely No Sinister Foreshadowing

"...when someone asked outright whether they were a man or a woman, the Friend replied with another Bible quote: "I am that I am." The speaker behind this quote in the Bible was God, suggesting that maybe the Friend was claiming to be God, made flesh. This would cause some trouble later on."

Ya think?

For now, though, enthusiastic followers grew in number. Help the poor, said the Friend, and followers said, "Yeah, that sounds right." Oppose slavery, said the Friend, and followers said, "Right on." Stay celibate, said the Friend, and followers said, "Hold on, let's not go crazy," and most ignored this advice and married. An exception: 50 women stayed single and formed a group within the movement known as the Faithful Sisterhood. If that name make them sound like militants willing to respond with violence when necessary, good instincts. Keep that thought in mind.

I've been told violence becomes more necessary the longer you go without sex.

4. The Murder Accusation

A bit too convoluted for me to quote just pieces of, but summary: someone tried to kill someone and the Public Universal Friend ended up being implicated (although they were in a whole nother state at the time), but not officially accused.

3. The Gore Commune

No, this isn't where the internet was invented or climate change became real.

At this point, we understand it sounds like this whole tale is building toward some kind of religious mass slaughter. So, let's warn you now that no such slaughter is forthcoming, but it's going to sound even more like one is, when you learn the Friends opened a commune known as The Gore.

Robbed!

With the Public Universal Friend having maybe tried killing someone according to the rumor mill, a mob assembled outside their home and rioted, pelting the place with "stones, bricks, bats, et cetera." There was no friendship to be found from the Quakers either. The church expelled two of the Wilkinson boys for joining the Revolution, expelled two of the Wilkinson girls for attending unauthorized religious meetings, and expelled Public Universal Friend for founding their own religion (this is a dealbreaker for most churches).

Yeah, they tend to dislike competition.

A little bit more time there revealed a new problem: Life in the wilderness sucked. These were people used to living in towns, and this commune lacked a whole lot of stuff, such as food. They set about growing crops – anyone raised Quaker back then had oat-based superpowers -- but it still wasn't a comfortable life.

Note to self: Learn how to live in the wilderness before going to live in the wilderness. (Don't worry; I won't do it - the wi-fi sucks out there.)

2. Arrested For Blasphemy

According to the dubious story, Public Friend had called people to watch them heal William's recently deceased daughter Susannah. Then someone, suspicious, asked to stick a blade into the corpse to make sure she really was dead. At this point, Susannah jumped up and fled the room, still wrapped in her funeral shroud.

Dubious or not, I'm totally going to use that in a story plot.

1. And The Verdict Is ...

Lewis dismissed the case, because, "What is this crap? Blasphemy isn't a crime. And if there were a law making it one, that would be unconstitutional." Even if the Puritans hadn't been so great about religious liberty, America had enshrined the freedom into law through the recently ratified Bill of Rights. Then Judge Lewis invited Public Friend to deliver a sermon to the court. Which might not be totally in line with the establishment clause as we know it, but hey, baby steps.

For some reason, I found this darkly hilarious.

And when Public Universal Friend did die, their congregation watched over the body, wondering if, maybe, they would resurrect once again. They did not. But still, just by living, they had proven that America is a place where you can be whoever you want. Religious founder? No problem. Neither male nor female? Sure, no one can stop you.

And there you have it: A story from history that I had never even heard the slightest whisper of. Certainly someone else has, but it was all new to me. It has pretty much everything you'd expect: death, betrayal, religious freedom, religious persecution, mistaken identity... everything but car chases and explosions, though to be fair there weren't a lot of those in the 18th century.

I'm just surprised no one's made a movie out of this yet. Or at least a Broadway musical.

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