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

November 1, 2020 at 12:02am

#997330

Happy Now?

Well, it's November now. Which means another round of 30DBC for me.

PROMPT November 1st

Write about something you want. Pick something that you don’t necessarily need, but would make you happy simply to have.

The first thing that comes to mind is a vaccine for... you know.

Not just for me, of course. I'm selfish, but I'm not that selfish. There are a lot of people suffering from this, both by having it and by having friends and family go through it, not to mention lost jobs and broken social networks. I'm fortunate for many reasons, but one of them is that no one I know in real life is sick... but then, I don't know too many people in real life anymore.

For myself, it might mean being able to travel again. I don't know if the foray I made to Alexandria last week was a good thing or not. I mean, I had a good time overall (except for that goddamned broken vending machine), but in a way, that just makes it worse. Winter is coming, as the saying goes, and with it an end to being able to sit on a patio and dine and drink beer in relative safety. More people congregating indoors, masked or not, means more chances for getting sick. I'm probably going to have to go back into hermit mode for a few months, which does not make me happy.

Whether travel in itself would make me "happy" or not is an open question, though.

I pretty much have everything I need and most of what I want (that flying car continues to elude me), and I'm self-aware enough to know that "stuff" doesn't make me happy; at best, it relieves the tedium of day-to-day existence. Hell, at this point, it might make me happier to dispose of a good bit of "stuff." Not to the point of going all minimalist or anything, but stuff does tend to accumulate.

But I don't think of happiness as a goal to strive for. I mean, sure, there's that line in the Declaration of Independence, that pesky "pursuit of happiness" thing, but I have other motivations. Besides, I'm pretty sure that the word had different connotations

250 years ago. I find that happiness is elusive in pursuit of it for its own sake; rather, it's a byproduct of other activities.

And I'm pretty simple, when it comes down to it. A good beer, some good music, a good book (all, of course, dependent upon my own subjective determination of "good") will do it. As will a few other things, like after I'm sick or in pain and suddenly I realize that I'm not; it's really my baseline physiological state, but feeling bad just emphasizes how good feeling ordinary is.

Hell, I don't even like the word. "Happy." It comes from the same Old English root as the word "happen." As in, I see it as something that happens to you, like winning the lottery (which might or might not make one happy). "Hap," apparently, did have the denotation of luck or fortune - both good and bad. The two French words I know for the equivalent concept are: content(e) and heureu(x/se) -- the latter of which also has connotations of good luck or fortune, and the former of which it's so far impossible for me to read without thinking of the equivalent word in English, which is not exactly a synonym for "happy," but connotes (to me, anyway) a more general, less exuberant feeling of wellness.

So I don't know if owning a flying car would actually make me happy or not, but I'm willing to take the chance.

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