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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October 1, 2024 at 4:27am

#1077543

Lyon Sleeps Tonight

Another kind of slow day yesterday, partly because of another run on the TGV. This one took me from Dijon south to Lyon.

I should note here, as a bit of a warning to other Americans who want to visit France: while many people, especially in the travel and hospitality industry, speak some English, and many signs are multilingual, the vast majority of the train announcements and postings are solely in French. While I have some facility reading the language, it's not as easy for me to follow the spoken language (yet).

To be fair, I can barely understand train announcements in English. Ever been on the NYC subway system? The announcements there only add to the confusion.

The hotel that the travel agent arranged for me in Lyon is a step up from the others I've been in here. Five-star instead of three or four. For this particular hotel, part of the upscale amenities is a really remarkably well-stocked bar on the top floor.

Naturally, it became incredibly important for me to check out the bar.

And oh, man, was it worth it. While many of the spirits are also available in the US, they're a bit cheaper here. Even the Scotch, so it's not just because of the EU. The menu featured several interesting-looking cocktails, too, some of which I, of course, had to try. I was the only one sitting at the actual bar, and the bartender did the whole bartender-showing-off thing with the flipping behind his back and squirting drinks from an arm's length away and juggling shakers and all that. If I didn't know better, I'd think he was flirting with me. Or at least angling for a tip, which really isn't done here, from what I've been told.

They also have an 18-year Yamazaki (Japanese scotch-style whiskey), which I've had in the US, but only rarely. I restrained myself, though; I'll be here for a few days, and I'll have other opportunities to drink the really expensive stuff.

With all that, I haven't yet had much opportunity to explore Lyon, which is a really quite large city compared to all the others (except Paris). Today isn't looking good, either, with rain in the forecast, but I have a food-tasting tour lined up for later. That's for tomorrow's entry, though.

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