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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10 Most Recent Blog Comments:

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').

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

1 Like

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

February 22, 2023 at 9:27am

#1045329

Lost

Another entry for "Journalistic Intentions"

[18+], based on this photo prompt: Maybe I'm Lost

I never get lost.

You probably hear that from men a lot, don't you? Or maybe you're a man and you say it to save face, while privately acknowledging, yep, I'm all turned around and I need to penis a way out of this, preferably without asking anyone for directions.

But for me, it's true. This is because I have a superior sense of direction, maps, a GPS, and a backup GPS. Even before GPS, though, I could always find my way out of a wrong turn.

Now, I may be cheating, because I rarely go into the woods alone. The woods are a classic place to get lost in, because it's hard to estimate direction by looking at a sky that you can't see through the canopy. The old bit of "common sense" about moss only growing on the north side of trees is bullshit, incidentally; one more reason why common sense is neither.

I have to say, though, that sometimes I wander around on purpose, and an outside observer might conclude that I'm lost. This has happened several times, like when I got pulled over in Texas for having Virginia plates (okay, I was also speeding, so I deserved it, but I want to emphasize that this was right after I ate the dust of several pickups with Texas tags who blew by me). The cop, who ended up issuing me a warning, asked me: "Are you lost?" Well, while I admittedly didn't know precisely where I was, having never been to North Texas before at the time, I knew exactly where I was going: Vernon, where I would end up eating the absolute worst steak of my entire existence, worse even than the ones my mom would turn into hockey pucks.

So, for anyone counting, that's two mis-steaks in one day.

I do it walking, too. Mostly in NYC, which is a remarkably large city in terms of area as well as population. Most of Manhattan's pretty easy to figure out, because of the street grid, but the older area to the southwest can get pretty mazey. Also, there are four other boroughs. "But Waltz, aren't you afraid of all the crime?" No.

How do I do it, this "not getting lost" thing? Well, remember, my spirit animal is a turkey vulture, and turkey vultures are birds, and birds never get lost. Sometimes they'll wander around looking lost (or in the case of vultures, circle around in an updraft), but their bird's-eye view keeps them oriented. Some of them even have little magnets in their head so they always know which way is north. Once you know which way is north, you always know which way is south, east, west, up, and down.

But more importantly, they can get up high enough to see the map. I can't fly, dammit, but I do run around with a little map in my head. As I noted a couple of entries ago, my memory is generally crap—but I'm good with maps. Like, one time, I was chatting with a blackjack dealer in Vegas. She had an accent, so I asked her where she'd come from. "Eritrea," she said. "You probably don't know it."

I said, "East Africa, next to Ethiopia."

Not saying this sense of mine is perfect, you know. None of our senses are. But I get by.

As for the bird in the picture prompt, sure, maybe it's lost.

And maybe it wants to be. After all, if it wanted to be unlost, it'd fly.

I just have to add, as an aside, that this photo is pretty awesome. You might have heard of the rule of thirds

in photography composition. This photo subverts it, and to good effect. While the main subject is close to the lower right compositional point, the bulk of it is in the bottom square, which is almost as big a no-no as centering it in the image. And yet, I find it well-framed. I'd have either gone for the bird's eye at the composition point, or the bird's center of gravity. Instead, what we get is, as near as I can tell, the graceful neck, which is also the only pronounced curve in the image. Contrasting that, most of the lines in the photo, the grasses, are vertical or near-vertical—with one notable exception, which is the blade right in front of the bird, providing counterpoint (I keep calling it a bird because a. it's a bird and b. I think it's a swan but I'm not entirely sure).

Yeah, I don't criticize photos much in here (though I'm probably more qualified to do so than most of the shit I criticize), but this one struck me as worthy of comment.

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