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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 13 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 15 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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').

1 Like

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.

April 10, 2022 at 12:01am

#1030456

Suffer the Little Children

Wading into sensitive waters today...

Ministry that once nourished Duggar family’s faith falls from grace
Scandal brought “19 Kids and Counting” to an end, coinciding with the waning influence of the Institute in Basic Life Principles.

I suppose you can call it a ministry. I call it a cult.

When the Duggar family caught the public’s eye in 2004 with its first television special, “14 Children and Pregnant Again,” the lifestyle depicted on screen was alien to many viewers.

As it should be. I suppose I should disclose that I never watched the show in any of its incarnations, so I'm not going to comment on its content.

Yet some Americans recognized their own values in the fundamentalist Christian family, as well as the influence of one man: Bill Gothard, founder of the Institute in Basic Life Principles.

One defining aspect of a cult is a single charismatic leader. The Catholic Church has a single leader, too, but he's not all that charismatic.

But today, the IBLP is losing income, and Gothard, 87, has been forced out over allegations that he abused young women working at its headquarters.

Which is another defining feature of a cult.

And the Duggars’ shows have been canceled: “19 Kids and Counting” in 2015 amid revelations that their eldest son molested four of his sisters as a young teen and “Counting On” last year ahead of his child pornography trial and eventual conviction.

No one seems to want to draw the obvious conclusion, connecting the dots between being a cult member and practicing deviant sexual behavior. And I can't say for certain that A is the direct cause of B, but I can say that one should consider the possibility and perhaps investigate that connection.

The Institute in Basic Life Principles began in 1961 as seminars by Gothard, an evangelical minister from Illinois with a master’s degree in Christian education.

Over 30 hours, he taught attendees how to lead successful lives by following his interpretation of Biblical principles and warned them away from television, popular music, alcohol, dating and public schools.

Well, I kind of agree with the television bit. But that makes it all the more hypocritical that the thing that drew this family into the public eye was being on television.

At the heart of Gothard’s teachings was the importance of respecting “God-given authority.” He preached a strict hierarchy of divine authority, with Jesus at the top followed by church elders, employers and husbands, who are responsible for protecting their wives and children below them.

And that right there sums up a big part of everything that's wrong with society right now, and has been for a long time.

In marriage, a man’s role is to provide “servant leadership” while “the woman responds with reverent submission and assistance,” preached Gothard, who has never married.

I will note here, without condoning or condemning, that while Catholic priests are famously forbidden to marry, Jewish rabbis pretty much have to be married. Because, the reasoning goes, how can one give marital advice as a spiritual leader without any personal experience therewith?

The Duggars’ wholesome image made them a poster family for the IBLP as their TLC show “19 Kids and Counting” became a ratings hit: Elder daughter Jill Duggar’s two-hour wedding special in 2014 drew an audience of over 4.4 million.

Again the disconnect between prohibiting television while participating in its worst excesses.

The IBLP said in a statement that it had “no comment with respect to the claims alleged,” because of a confidentiality order. Gothard has since denied any wrongdoing. He declined to comment when reached by phone.

I'm also not going to comment on whether Gothard is guilty of those particular charges or not. Not my department. But in a hypothetical situation where a charismatic cult leader gets accused of inappropriate behavior from multiple sources, I'm inclined to take the word of the accusers at face value.

A year after Gothard resigned, the Duggars had their own fall from grace. In May 2015, a leaked police report from 2006 revealed that the eldest Duggar son, Josh, had been accused of molesting five girls as a teenager.

While that never led to conviction, apparently the guy was later convicted on other charges.

Barr, the Baylor University history professor, said Gothard’s fundamentalist patriarchal teachings help create environments where abuse is more likely to go unchecked.

“As a man, they have more authority than women,” she said. “Their voice counts more, and women have this propensity to be valued more for their sexual role, so it allows abuse to flourish.”

Okay, so maybe someone is trying to connect the dots. Still, to be fair, that's not exactly scientific evidence.

And also to be fair, they claim to preach Biblical principles, and I don't recall anything in the Bible forbidding child porn. So I guess that, to them, they're still on the right side of morality.

I just disagree with their definition of morality.

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