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.

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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.

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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.

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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.

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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 7, 2021 at 12:03am

#1016946

My Baloney Has a First Name...

Today's article is more about reading than about my comments. But of course, I have comments, too.

The Baloney Detection Kit: Carl Sagan’s Rules for Bullshit-Busting and Critical Thinking
Necessary cognitive fortification against propaganda, pseudoscience, and general falsehood.

This article is over seven years old. I only wish more people had read it before... well. You know.

Carl Sagan (November 9, 1934–December 20, 1996) was many things — a cosmic sage, voracious reader, hopeless romantic, and brilliant philosopher. But above all, he endures as our era’s greatest patron saint of reason and critical thinking, a master of the vital balance between skepticism and openness.

I never read much of his stuff, and only saw a few of his videos, but I guess you could say he was an inspiration for me. I wouldn't call him a saint, though. It's not a good idea to elevate people like that. Besides, have you seen what passes for a saint lately? No thanks.

In a chapter titled “The Fine Art of Baloney Detection,” Sagan reflects on the many types of deception to which we’re susceptible — from psychics to religious zealotry to paid product endorsements by scientists, which he held in especially low regard, noting that they “betray contempt for the intelligence of their customers” and “introduce an insidious corruption of popular attitudes about scientific objectivity.”

At the same time, though, scientists have to make money somehow, and they're certainly not going to do it by stashing mircochips in vaccines or hoaxing us into believing the planet is round and global warming is an existential hazard.

Because they're not doing either of these things.

Through their training, scientists are equipped with what Sagan calls a “baloney detection kit” — a set of cognitive tools and techniques that fortify the mind against penetration by falsehoods:

Ideally, yes. In practice, some slip through the cracks. But one doesn't have to be a scientist to use some of the tools.

The article goes on to list the actual thinking methods involved. I've highlighted a few of them in here before.

Just as important as learning these helpful tools, however, is unlearning and avoiding the most common pitfalls of common sense.

I've come to the conclusion that common sense is neither. It's often wracked with nonsense, and everyone's idea of "common" is different. Science tries to get past that and into some sort of objectivity. Often, its conclusions defy common sense, and yet are still true, to the extent that anything can be called "true."

I'd encourage everyone to read the next section, because it lists a bunch of fallacies that, in order to avoid, one must recognize. They're important, and I know I need to be reminded of them repeatedly for some of them to sink in.

The Demon-Haunted World is a timelessly fantastic read in its entirety, timelier than ever in a great many ways amidst our present media landscape of propaganda, pseudoscience, and various commercial motives.

And one of these days, I really should read that.

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