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

March 13, 2021 at 12:01am

#1006274

Crow Magnon

We've all known for a while that corvids are smart (and smart-asses), but it's always good to have some science backing things up.

Brainiacs, not birdbrains: Crows possess higher intelligence long thought a primarily human attribute

As usual, I have a few quibbles with the article, but I think it's generally effective in communicating what's going on.

Whether crows, ravens, and other “corvids” are making multipart tools like hooked sticks to reach grubs, solving geometry puzzles made famous by Aesop, or nudging a clueless hedgehog across a highway before it becomes roadkill, they have long impressed scientists with their intelligence and creativity.

Don't know why they had to put the quotes there. And it's not only scientists who are aware of these bird's big brains -- anyone who's watched them long enough can tell you they're scary smart.

Research unveiled on Thursday in Science finds that crows know what they know and can ponder the content of their own minds, a manifestation of higher intelligence and analytical thought long believed the sole province of humans and a few other higher mammals.

1) This article is dated last September. "Thursday" doesn't help much now.

2) I've known some humans who can't "ponder the content of their own minds."

3) The author uses the adjective "higher" here and in a couple other places. I don't like it. "Higher" implies a hierarchy that can be misleading, or it could mean "more intelligent," in which case it's redundant in an article talking about intelligence. No species is really more advanced than any other, in general; they've merely evolved different survival strategies, only one of which is what we'd call "intelligence."

Knowing what you know is also a form of consciousness, and the discovery that more and more nonhumans seem to have it raises tricky questions about how we treat them.

Unless they're also delicious, like pigs.

The article goes on to describe, in detail, an experiment that demonstrated the cognitive abilities of crows, and another that studied the neuroanatomy of sky rats. Er, I mean, pigeons. I have to admit I kinda zoned out on this part, more interested in the conclusions than the methodology.

“Besides crows, this kind of neurobiological evidence for sensory consciousness only exists in humans and macaque monkeys.”

Snort. He said "macaque monkeys." Snicker.

“In theory, any brain that has a large number of neurons connected into associative circuitry … could be expected to add flexibility and complexity to behavior,” said Herculano-Houzel. “That is my favorite operational definition of intelligence: behavioral flexibility.”

I do wonder about this, but the woman quoted there is a scientist and I'm just an onlooker. Either way, though, I find this sort of thing fascinating. It's like how a crow can't resist a shiny thing, or how a raven can't resist annoying an angsty poet.

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