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🎶

about 22 hours 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 25, 2024 at 12:13pm

#1074374

Let Me Give You a Hand

Wrapping up this round of "Journalistic Intentions"

[18+] with the final randomly-selected prompt:

Antwerp

And seriously, I used a random number generator. It's pure chance that it landed on a city in Belgium at a time when I'm planning a trip to France and Belgium.

And I'd love to visit, really, I would. The city hosts the Bollekesfeest every year, which features local beer; if you need to know anything about me at all, it's that Belgian beer is my Kryptonite. Sadly, the event takes place in August, and the timing doesn't work out for me. And it's home to the Antwerpse Brouw Compagnie, which I've heard brews a beer called Seef or Seefbier whose origin dates back to the 16th century. For context, that was before European colonization of the Americas. Hopefully the beer itself isn't that old, but damn, it sounds like something I need to try. (Their more modern brews look awesome, too.)

The problem is I don't have unlimited time. Tant de bières, si peu de temps. Yes, that's French, though Antwerp is in the Dutch-speaking part of the country. Hell, it's practically Nederland, another country I'd love to visit. My primary interest in Belgium this trip is their famous abbey breweries. Monk beer. So, Brussels and maybe Bruges. I think. Yes, I'm aware that Brussels and Antwerp are quite close together, maybe 40km; I'm perfectly capable of reading a map. There's still not enough time to do everything.

Besides, I gotta save some cool stuff for the next trip.

And no, I still haven't picked up much of the language. I tried. It's harder than French, because it's simultaneously very similar to English and very, very different. All of the Dutch-speaking people I know have perfect English, and it's embarrassing that I can't seem to return the courtesy. Well, maybe not "perfect" English (because such a thing does not exist), but still better than that of most Americans.

As you know if you've been following along, one of my other interests is etymology, and the origin of the name "Antwerp" is debatable. But, like Rome, in addition to the actual etymology, it has a mythological etymology, and it's glorious.

I'll just let Eupedia

relate it:

The legend says that the name 'Antwerp' comes from the Dutch "hand werpen" meaning "hand thrown".

The story has it that a mean and nasty giant controlled the river traffic, demanding exorbitant tolls. Those who refused to pay had one of their hand cut off.

But one day, a young and brave Roman soldier named Silvius Brabo managed to kill the giant. He cut the giant's hand and threw it in the Scheldt River, giving the city its name.

I need details, dammit. How did Silvius Brabo defeat the giant? More importantly, how did he keep the giant from landing on him when he killed it? I did find that the giant's name was Antigoon, which somehow doesn't mean "he who fights against goons." And is this the first example of classic rent-seeking

behavior and its inevitable consequences?

Sadly, the likely actual origin of the name is far less interesting: "The name comes more probably from the word ‘aanwerp’ (alluvial mound’), which describes the first settlement's geographical feature." (Same Eupedia source as above.) Yeah, apparently "werp" served double duty, much as its English descendant "warp" does. Though one could make the argument that the alluvial soil was "thrown" there by the river, I suppose.

And now I want to visit Antwerp more than ever. Dammit. I don't have infinite funds, either. Maybe I can start charging tolls.

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