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

July 4, 2023 at 12:33am

#1052113

No Fly Zone

Clearly, it's sorcery. Or maybe ghosts.

No One Can Explain Why Planes Stay in the Air
Do recent explanations solve the mysteries of aerodynamic lift?

Actually, what "no one can explain" is why I fell for the clickbait headline. Maybe because I expected better from Scientific American.

In December 2003, to commemorate the 100th anniversary of the first flight of the Wright brothers, the New York Times ran a story entitled “Staying Aloft; What Does Keep Them Up There?”

One could say it doesn't much matter, because it's obvious that they do stay up. The vast majority of the time, anyway.

But that's not science. This is like the old nonsense about how bumblebees can't fly (spoiler: they actually can).

To answer it, the Times turned to John D. Anderson, Jr., curator of aerodynamics at the National Air and Space Museum and author of several textbooks in the field.

Okay, I can accept those credentials, for now.

What Anderson said, however, is that there is actually no agreement on what generates the aerodynamic force known as lift. “There is no simple one-liner answer to this,” he told the Times.

"There is no agreement" is not the same thing as "there is no simple answer."

In the interest of full disclosure, I had, for a long time, been under the impression that it's all about Bernoulli's Principle (airfoil causes air above the wing to have higher velocity and thus lower pressure), though it never really seemed to explain everything. And it doesn't. For instance, you know those old planes that could fly upside down? Can't be explained by airfoils alone.

By far the most popular explanation of lift is Bernoulli’s theorem, a principle identified by Swiss mathematician Daniel Bernoulli in his 1738 treatise, Hydrodynamica.

Told you so.

The article goes into this in more depth, of course, but I won't reiterate much of it here. Basically, there are other factors involved as well.

The controversy, as far as it goes, seems to be about how much each of those different factors contribute. There's also unstable, chaotic effects at play, which are poorly understood right now.

But that's not why I linked the article. I mean, sure, this stuff's fascinating to me, but I don't expect everyone to be riveted by the article or whatever. No, my main points here are to show that a) just because you don't fully understand something doesn't mean it's not real; and b) I'm really annoyed that SciAm would use a headline like that. It fosters distrust in science, something that's pervasive enough in the world right now. "Why should I trust scientists?" I can hear the willfully ignorant saying while clutching a Coors Light. "They can't even explain how airplanes fly!"

It's true that science sometimes gets things wrong. I'm not contesting that.

It is even more true that without it, we'd get a whole lot more wrong.

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