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.

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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.

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.

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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

October 5, 2024 at 12:55am

#1077785

It Was Just Waffle

Even on the TGV, it takes a while to get from Lyon to Brussels. Worst, Brussels wasn't my actual destination; I had to hop on an intercity train to Bruges. Those trains are open seating and run frequently; they're more like NYC subways than what I'd expect from intercity (IC) trains, only without the piss, buskers, and giant Saks sacks.

Well, I guess technically, they're more like NJT trains, but more people know the subway.

Anyway, I was concerned about time, because I had to be at a certain statue at 6:30 pm or I'd miss a beer tour (and of course I couldn't let that happen under any circumstances). So I ran to catch an earlier IC train, which I barely did because the TGV was late getting to Brussels. Then, once in Bruges, I had to get to the hotel even though the meeting place was kind of between the station and the hotel, because who schleps travel bags on a beer tour?

You'd think I'd have learned my lesson back in Tours, but, apparently, we ugly Americans just won't take a hint. Once again, the hotel was 1600 meters from the train station. Once again, the terrain between was mostly flat. And once again, I cursed every single goddamn cobblestone in Europe.

I got to the hotel at 6, exhausted. My left knee is still wonky, and so is my right ankle. Of course it can't be one leg; it's gotta be both so I can limp with every step instead of just half of them.

Now, the hotel is older than European colonization of America (unless you count the Norse incursion, which I don't because they didn't make a lasting colony), so it has a few... interesting features. One of them is an abundance of stairs. Another is that, in order to get to my room, I had to take an elevator up one floor (or carry bags up a spiral staircase, which, well, no) then walk down a hallway, up a ramp, bounce up two steps, take a few twists and turns, and then get on a separate elevator to take me up half a floor.

At least the hotel has been retrofitted with electricity.

I barely had time to throw the bags in my room before I had to turn around and make my way to the statue. One of the streets on the way was called Steenstraat. Now, I don't know very much Dutch, I'm afraid, but "steen" and "straat" are two words that I am familiar with: Stone Street.

Even without bags, cobblestones are a pain in the ass.

When I got there, the guide was waiting impatiently, tapping his foot. Apparently, he'd messaged me to meet at 6 instead of 6:30. I'd never gotten any such message. Fortunately, his English was excellent, and I showed him that no, I'd received nothing. Unfortunately, the reason he'd wanted to meet earlier was that the best beer places closed at 7, and now we wouldn't have enough time.

All that running and stumbling on cobblestones... all of that... for nothing.

Well, not completely nothing. We agreed to try again for Sunday at 2:30. And the guide pointed me at a nearby restaurant which he said was good. Now, I'm pretty sure it's the law in Belgium for every restaurant to be good, but in this case, he was definitely right. I managed to hit the entire Belgian flag

in one dinner: Beer, chocolate, and waffles. And meatballs for the main course, but meatballs aren't on the flag. All, incidentally, delivered to the table by a Japanese nekorobot . No, seriously. This thing had cat ears and a screen with an animated cat face. I felt like I was in a Black Mirror episode. Or at least Doctor Who. I should write a book: Cobblestones and Advanced Robotics: A Study In Contrasts.

Oddly enough, even though the beer I had was very local to Bruges, I'd had it before; somehow, a keg had made its way to my hometown in 2021, and, as I keep meticulous beer-drinking records (okay, it's an app), I was able to determine that I'd sampled it back then. Of course, I didn't mind drinking it again, and there are plenty of other beers here that never make it out of the EU, so it's not like I wasted the trip.

I could have done without all the quaint fucking cobblestones, though.

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