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.

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

January 19, 2021 at 12:01am

#1002398

Where We're Going, We Don't Need Plans

PROMPT January 19th

Do you like things to be carefully planned or do you prefer to just go with the flow? Do you get upset easily when your plans change unexpectedly or for reasons beyond your control? Imagine you are taking a road trip - how much of the trip do you plan in advance?

Oh yes, please, by all means, make me imagine doing one of my favorite activities during a time when I effectively can't. That won't irritate me in the slightest.

/sarcasm

I mean, sure, I could take a road trip. Technically, there's nothing stopping me from getting in the car and driving. It's just that a lot of the reasons for me to take a trip -- restaurants, bars, and breweries -- are closed, have limited hours, are outdoor-only and it's winter, or are simply a Bad Idea during a pandemic. So there's not much point.

Anyway, I'm predicting that not too many people are on either extreme of the planning/pantsing scale where this is concerned. Like most things, it's a spectrum, and most people fall somewhere in the middle.

It is difficult, if not impossible, to plan everything down to the last minute. You get traffic jams, unexpected detours, closures, squirrels (meaning, like, you're concentrating on doing something and then you see something shiny and you're like Dug the dog from Up going, "SQUIRREL!"), etc. It's equally unlikely to be able to have no plans whatsoever; at a bare minimum, your plan is, "I'm just going to tag along with this other person and do whatever they want to do." I mean, sure, technically maybe it's possible to get on the road and choose a path at random whenever you get to an intersection, but even planning to do that is a plan of sorts.

I haven't done that, exactly (though now that I think of it I might have to try it sometime), but one of my favorite ways to take a road trip is to choose a destination literally at random - I once found a site that would generate random coordinates, though I suspect it's based on latitude/longitude, which makes destinations toward the poles somewhat more likely than destinations toward the equator (think about it - it's because lines of longitude converge, so you have a denser array of possible points in, say, Canada than you do in, for instance, Mexico, while on the flip side, there's literally nowhere to actually go in Canada once you get out of southern Ontario).

So it's not as random as I'd like, but keeping it to the US alone, it's close enough for what I want to do.

Oh, incidentally, sometimes you hear about people throwing darts at a map. That's semi-random, but the map is a projection, so again, some areas are more likely than others. If you think about it, throwing a dart at a globe wouldn't work either. I still haven't worked out a way to get a truly random location, with each point equally likely, on a round planet.

Another thing I found was an app that generates a random zip code. Twice, I've rolled up a zip code and headed there; fortunately, both areas were in the Northeast: one in NJ and one in Massachusetts. Random coordinates, on the other hand, have put me in places like Montana, Alabama, and the actual middle of actual nowhere in central Nevada. To name but a few.

The other downside of random coordinates is that I have to ignore any that are generated in large bodies of water. Other people might follow their GPS into a lake, but I'm not that stupid. (Don't blame the technology. It's always the driver's fault.) The obvious downside to using zip codes is that some have much bigger areas than others.

I said "close enough for what I want to do," but what I want to do is visit breweries and see whatever sights are near (or on the way to or from) these random destinations. Because I take very literally the maxim that "it's the journey, not the destination, that matters." I mean, sure, there are times I care where I end up, like when I'm visiting someone or decide there's something in particular I want to see, but for the most part, I just want to see and experience everything I can. I would even say that there are no destinations; there are only stops on the journey.

So to address the second question, no, obviously I don't usually get upset when the plans change. Normally, I see it as just another part of the adventure. There are exceptions, like when I've made plans with other people and something happens that makes me inconvenience them (for instance, a cancelled flight or heavy traffic delays). But for me alone, nah, give me something new and interesting.

I should probably go ahead and "plan" my next road trip, by which I mean pick a few random destinations and research nearby breweries. Or take a few WDCers up on their offers to meet at various locations. Problem is, I still don't know when I'll be able to do it, and it makes a big difference whether I'll be able to go in June or have to wait until next winter. There are roads that close down completely in the winter, especially out west, and there's always the chance of getting stuck in sn*w. I once thought I'd avoid this by doing a winter road trip through the southern part of the US -- and ended up in a blizzard all the way from Winslow, Arizona to Amarillo, Texas. No one told me there were blizzards in goddamn New Mexico.

And really, my next trip is probably going to be to Belgium, which was supposed to happen last year. That one's going to take a fair bit of planning, but I'm still leaving room in the plans for squirrels.

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