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 6, 2020 at 12:03am

#995149

Road Hazard

For today, my second entry chosen at random from the prompts in

Journalistic Intentions

(18+)
This is for the journal keeping types that come to PLAY! New round starts July 1!
#2213121 by Turkey DrumStik

Hazard Ave.

Another Street Sign prompt... I told you I pick these things at random. My only rule is not to do one twice.

Even though I designed roads, I never paid much attention to the general distinction between streets, avenues, ways, lanes, drives, etc. I mean, I had this vague notion that a street is urban while a road is rural; a court is a short street that either loops or ends in a cul-de-sac, and an avenue is a particularly broad street. But I could name exceptions to each of these general rules.

I just designed the things. I never got to name them. This is probably a good thing, because then there'd be a Thunder Road, a Back Street, an Icant Drive, and an Ivelostmy Way in my town. To name but a few.

And really, some of these naming conventions are purely arbitrary. The street I live on is a court, because it ends in a cul-de-sac; a nearby street is a drive, even though it's a loop. Another nearby street is designated an avenue, even though the closest part to me is actually narrower than the court I live on (later on, it's divided by a tree-planted median). The street=urban and road=rural is more of a design specification; streets generally -- again, you can find exceptions very easily -- include curbs and sidewalks, while roads (including highways and freeways, another blurred distinction) do not. The drainage systems are completely different for streets and roads, and from a design standpoint that's what matters.

As I'm learning French, it seems they make the same distinction: a rue is an urban street, while a route is a rural road. Meanwhile, in America, we can't decide whether to pronounce "route" like root or like rout. (In French, it's closer to root.)

Speaking of France, the Platonic ideal and archetype of the "avenue" is the Champs-Élysées in Paris, formally the Avenue des Champs-Élysées, because yes, avenue is a French word. And it's only very recently that it finally dawned on me that the name of this prototypical avenue translates as Elysian Fields. I mean, I could have looked that shit up if I'd been arsed to do so, but as usual, I was not. Hell, it took me fifty years just to learn how to pronounce it correctly so as not to get smacked by a passing French person.

Hopping back across the pond, let's look at New York. It has avenues, too. In Manhattan, avenues run north-south (actually more northeast-southwest, but whatevs) while streets run perpendicular to them. That is, on most of the island. The lower and older part is confusing as hell. The Dutch probably did that, back when it was New Amsterdam, specifically to annoy the hell out of the rigid English when the latter acquired the city from the former. Or, well, no; there was no real planning involved, but that's my headcanon.

Anyway, the point is, the avenues of New York are merely broad streets, and most of them are one-way. You think of an avenue, you probably think tree-lined, probably has a median, otherwise looks like an airport runway. Oh, sure, there are trees, but almost all streets in New York have trees. Pretty sure it's the law there. You look at a picture of the Champs-Élysées, though, and it's positively teeming avec les arbres.

Which reminds me of a joke:

Why are the streets of Paris lined with trees?

Because Germans like to march in the shade.

There is one notable exception to the rigid grid structure of middle Manhattan, and that's arguably the second-most famous street in the world: Broadway. (I'm sure that as an American I should proclaim it is the number one most famous street in the world, but come on, just look at a picture of the Champs-Élysées. That sucker is just plain beautiful.) Broadway is so special that it doesn't have Street, Avenue (it would qualify as an avenue under most definitions of the word), Road, or any other description. And it's not Broad Way, or Broadway Way, because that would just be silly. It's just Broadway, and it cuts right across the otherwise regular grid pattern that defines Manhattan and makes it actually a rather easy city to drive in, as compared to, say, Boston or DC.

Fun Fact: Broadway starts in Lower Manhattan, but it runs across into the Bronx and on into Upstate New York, not stopping until after it's passed through Sleepy Hollow. Yes, that Sleepy Hollow, of Ichabod Crane fame and the setting of a TV series that started out so well and then went to shit. In fact, there's more to Broadway outside of Manhattan than there is in Manhattan. One of these days I want to follow it for its entire length. It's only something like 33 miles long; given Manhattan traffic it should only take me a week or so.

Point is, with all the traffic, trees, statues, medians, curbs, stoplights, signs, blinking lights, crosswalks, parking, pedestrians, and potholes you inevitably find on something called an "avenue," the only thing shocking about Hazard Ave. is that it's not a far more common name for a street.

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