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.

2 Like

June 12, 2023 at 9:07am

#1050969

Swimming

Another one for "Journalistic Intentions"

[18+]...

Swimming

Ever wonder why you don't have fur?

Other apes are covered with their own pelts, rather than those of other animals. And yet, we humans, who manage to survive in some of the world's most extreme climate conditions, don't. Some are hairier than others, of course, and evolution played an especially big joke on men, who, with age, tend to lose the hair on our heads while gaining it in our ears and noses. (Not me, of course. But some other men.)

One hypothesis that was presented a while back is called the "aquatic ape hypothesis." Noting that most aquaphilic mammals are mostly hairless, the idea is that, at some point after diverging from our common ancestry with chimpanzees around six million years ago, some proto-humans spent a lot of time swimming.

Fur is generally not suited to swimming. Drag, I suppose. The obvious examples are whales, seals, and any number of other aquatic mammals. The same can be said for feathers; hence, penguins. And then, by this theory, we left the water and yet maintained an affinity for it, as illustrated by the popularity of swimming.

There are, of course, counterexamples, such as beavers. Maybe polar bears.

The problem with the aquatic ape hypothesis, however attractive it may be, is that it's wrong.

Well. Probably.

Part of the problem is a misunderstanding of how evolution works. Another problem is the old chicken/egg thing: did we become (relatively) hairless because we were swimming, or did we get (relatively) good at swimming because we were hairless?

There's little doubt in my mind that lack of fur makes for better swimming (not to mention drying off after), but there are several other possible reasons for the adaptation. Including that it's not a true survival adaptation at all, but a side effect of other genetic changes, or a result of sexual selection, or some combination of factors.

What we do know is that, whatever the evolutionary reason, humans have the capacity to learn how to swim, and many do, and derive great pleasure from it, or even from watching other people swim. Not me, of course (though I did learn as a child and can at least still dog-paddle), but other people. And it can provide a survival advantage, especially when you get thrown off the boat for being too pedantic.

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