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🎶

about an hour 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.

2 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').

1 Like

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 24, 2020 at 12:01am

#999003

Serenity Now

"Peace" is not actually an uncommon state of being for me.

PROMPT November 24th

Think back to a time when you felt completely at peace. What made you feel that way?

While much is made of "happiness," I find it unsatisfactory. I've ranted in here about that before, I'm pretty sure. But peace? That's a different feeling entirely, in my opinion. When I think of happiness, I think of something boisterous and gaudy, like a river rapids; peace, on the other hand, is a calm lake.

Both have their place, of course.

Some find that feeling of calm to be boring, I suppose. I do, too, after a while. It's not a state of mind I'd want to hold onto for an extended period. You forget about the things you're supposed to be doing, which is desirable in the short run, but then shit catches up to you and boom, turbulence.

In the Before Time, I'd visit breweries to sample their wares. I've done that a bit since the pandemic started, but mostly just ones that are nearby; with lockdown rules, it's not as straightforward as it was. At the best of times -- when the place isn't very busy and when I don't have to worry about being anywhere at any certain time -- I'd achieve a feeling that I like to call "beerenity." Not drunk, just... the calm knowledge that everything is right with the world (even though it never really is, at least one can pretend it's so, briefly, for the sake of one's sanity).

That feeling is one of peace.

I don't mean to give the impression that I need to drink to achieve this state. There have been plenty of times when I've reached it completely sober, usually after finishing something difficult, or exercising to physical exhaustion. Or while driving, which certainly doesn't happen when I've been drinking. With beer or other libations, it can be a delicate balance; drinking too much leads to mere mental fog or drunkenness. Which, I should hasten to add, in my case doesn't mean doing anything untoward, just that I can't enjoy it as much.

So I think the last time I felt truly at peace was a few months ago, and yes, beer was involved.

My state's lockdown kept my local taphouse closed for a while. This didn't stop me from getting delivery food, or drinks from local breweries and/or the grocery store, but one thing I'd enjoyed since the place first opened a couple of years ago was going to the taphouse, sitting at the bar, and either trying some new brew or indulging in an old favorite. It's just a different experience from merely drinking at home.

Anyway, and I think I wrote about it in here at the time, at some point back in June, I think -- late spring or early summer -- the taphouse reopened. The bar was unavailable (still is), and I didn't want to be inside anyway, but after a couple of months of stewing at home, I got to sit on the covered patio of the taphouse and order beer and some food. They have a sound system on the patio, so I just sat there enjoying the warm weather, drinking beer and listening to the music, and things felt almost normal. I would even say it felt... I can't think of the word right now. Promising? Hopeful? Auspicious?

Good, anyway. It felt good.

It didn't last, of course. It never does, nor would I expect it to. I think I ended up overindulging and staggering back home (I know I've mentioned in here before that the place is within easy walking distance, even after drinking) to pass out, but for a while there I'd achieved beerenity.

I've been going back once a week, even though the weather's gotten cooler so the patio isn't as welcoming. Enjoyable experience, every time, though never again as blissful as that first one after they reopened. Yesterday was one of those days, and it's looking like next week might work out also.

I'm going to be sore about not being able to go out to California this year. Still, somehow, I'll find peace.

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