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

October 4, 2024 at 1:59am

#1077735

Coats Do What Now?

For my final full day in France, I took a small-group tour to the wine region of the Côtes du Rhône.

In contrast with the previous day's trips, I wasn't very familiar with the Côtes du Rhône vintages. This, of course, is why I went. I find that, when traveling, some mixture of the familiar and the new is ideal, though I expect the proportions would differ for everyone.

While I don't think I'd ever had a wine from this region before, some of the grapes they're based on are fairly popular to grow in the US.

The tour group itself was pretty interesting. Some of us ugly Americans, a couple from Quebec, and three people from Australia, who, while initially distrustful of my Americanity, quickly warmed up when I assured them (out of earshot of the tour guide) that I very much enjoy Australian Shiraz. While the guide insisted that Syrah, one of the grapes in Côtes du Rhône, is not the same thing as Shiraz, it totally is; it's just grown on very nearly the exact opposite side of the world.

The tour guide himself, who looked a bit like Kiefer Sutherland, was otherwise very knowledgeable, and spoke good enough English that all of us varied Anglophones mostly understood him.

The problem comes in when some unknown place name, or grape name, pops up in spoken French.

Like, if someone comes up to me and says "Bonjour, peux-je vous aider?" I can work out what they're saying. But if I'm at a wine tasting, even if it's mostly in English, it might go something like this:

"This wine comes from the pl***du*** region, near B***on and the S**** river valley. If you visit there, be sure to go to D****de*****, a fine restaurant that serves bl*****, v*******de la *******tion, and many very good wines, including Petit ********tre, Vo******** Blanc, and sparkling Dr*******"

It's tiring, to be honest. I started asking for spellings. To make things worse, my hearing isn't that great thanks to 50 years of rock concerts (no regrets), and I'm not asking you to repeat yourself because I don't understand French; I'm asking you to repeat yourself because I literally can't hear you over the ambient mumbles.

Ah, well, soon I get to complain about the same thing in Dutch (or at least the version of it known as Flemish; I don't know the details there), as I'm leaving for Bruges today. It's a tight schedule: catch a train to Brussels, transfer to a train to Bruges, get to the hotel from there somehow (Uber seems to be wonky in Belgium; apparently, only taxi drivers can do Uber, thus defeating the whole purpose but providing job security), then find a certain statue in a city saturated with statues to meet a beer guide who will take me... well, I don't care, as long as there's Belgian beer.

I'll wrap up with another landscape picture, this one of the Rhône Valley.

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