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.

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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

October 1, 2022 at 12:02am

#1038403

Muddy Talks

I'm participating in "Journalistic Intentions"

[18+] again this month, so some of my entries will be prompts from that activity.

This is one of them—as usual, selected at random.

Locks, dams, and the future of the Mississippi

When I was in Minneapolis in the summer of 2021, I got to see some of these locks and dams up close. I wrote, briefly, about it at the time: "Dam It"

So apparently, based on the link above, there's talk about removing some of them. Not the ones I toured, but downstream from there.

You might think that, being a civil engineer whose whole thing was hydrology, I'd have an opinion on it. After all, I have opinions on plenty of things that I know far less about. But you'd be wrong. Well... I do actually have an opinion, which is "they should do what's expected to have the best overall outcome." Which is why we have people do studies for this sort of thing.

If I thought about it long enough, I'd probably come to the conclusion that this should probably be my opinion on everything else, too. But no, then life would be no fun, and worse, I'd have nothing to blog about.

Besides, rivers are complicated. They build dams for reasons: flood control, erosion prevention, power, recreational purposes; usually some combination of these things. There's always a trade-off too, like when the dam blocks fish spawning or whatever, or the resulting lake drowns a community.

Left to its own devices, a river will shift over time. Not just geological time, but often in one human lifetime. You can see the effects if you look closely at a map of the rest of the Mississippi: horseshoe bends where the river once flowed, and down the center of which a state line was surveyed, have gotten cut off, leaving, for example, a bit of Tennessee on the Arkansas side of where the river is now. Several bits, actually. Some of that might have been due to Army Corps projects, but probably there was also some non-human-caused shifting over the years. If you wanna see, look up Memphis on Google Maps and then scroll north.

And I'm not saying this is bad, or good; it just is. We modify rivers for our own purposes, and for a long time we didn't think much about the consequences. Now that they're modified, though, there are consequences to unmodifying them, as well.

Even further downstream, in Louisiana, the primary Mississippi discharge has been trying to shift over to the Atchafalaya river (no, I can't pronounce it, either). If it weren't for levees and locks and whatnot—engineering projects—it probably would have switched channels by now. From what I understand, it used to switch back and forth every thousand or two years. As one channel silts up, water flows to the other, and vice-versa.

This, of course, would leave New Orleans dry (if not high), so there are economic reasons not to allow it. But there are probably good solid ecological reasons to do so. But we're a part of the ecology, too, part of nature, and we have to make decisions based on incomplete information.

I'm just glad it's not my job.

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