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.

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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.

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June 17, 2024 at 7:47am

#1072773

I Want It To Work, But...

Today's article, from SciAm, is an ad. Though it's for a book, which I've repeatedly insisted is appropriate here (much as movie previews are appropriate ads before movies), I just wanted to state that up front.

Can Scientific Thinking Save the World?
A physicist, a philosopher and a psychologist are working together to bring better, smarter decision-making to the masses

As I've also repeated, my default answer to any binary headline question is "no."

I mean, I'd like it to be "yes," but the world stubbornly refuses to conform to my wishes.

Spurred by what they saw as a perilously rising tide of irrationality, misinformation and sociopolitical polarization, they teamed up in 2011 to create a multidisciplinary course at the University of California, Berkeley, with the modest goal of teaching undergraduate students how to think—more specifically, how to think like a scientist.

Seeing that now, 13 years later, that particular (red) tide keeps rising, I'd say the world stubbornly refuses to conform to their wishes, too.

Now the three researchers are bringing their message to the masses with a new book, Third Millennium Thinking: Creating Sense in a World of Nonsense.

Like I said: ad.

And their timing is impeccable: Our world seems to have only become more uncertain and complex since their course began, with cognitive biases and information overload all too easily clouding debates over high-stakes issues such as climate change, global pandemics, and the development and regulation of artificial intelligence.

Perhaps they're just pissing into the ocean.

Scientific American spoke with Perlmutter, Campbell and MacCoun about their work—and whether it’s wishful thinking to believe logic and evidence can save the world.

The rest of the adarticle is in interview format. I find it interesting, which shouldn't be surprising, given that the disciplines in question are ones I've discussed here on numerous occasions. But no point in rehashing points.

Instead, since the subhead up there cries out for it, I'll attempt to make up a joke:

A physicist, a philosopher and a psychologist walk into a bar. Bartender says, "What'll it be?"

The physicist says, "The answer depends on the level of our knowledge of all the prior states of the system."

The philosopher says, "The future is unknowable; we just have to wait to find out."

The psychologist says, "I'll have a beer. These two chucklefucks want wine."

...I didn't say it would be a good joke.

5 Like • Comment • View 4 comments

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