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.

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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.

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October 22, 2023 at 8:22am

#1057828

Revisited: "A Gift Beyond Price, Almost Free"

Today is a Sunday, on which it's been my habit, recently, to take another look at random older blog entries. This time, the dice landed on one from just under two years ago: "A Gift Beyond Price, Almost Free"

In it, I comment on a New Yorker article from the previous year, which in turn was a book review. The article

is still available as of today.

Now, usually, I go through and comment on things that have changed, or point out where I embarrassed myself with my comments. For the latter, I will admit to misspelling Rush drummer/lyricist Neil Peart's name in that entry. Apologies, Neil Peart's Ghost. Not that he read my blog even when he was alive.

For the former, I don't think much of anything has changed, which only supports my continued refrain in that entry every time the article points out where capitalism is exploitative: we obviously don't really care, because we don't do anything about it.

Even back in 2021, when I wrote the entry in question, I'm pretty sure I'd given up on humanity being able to take collective action for its own good. As if continued denial of climate change wasn't enough proof of that, we were in the midst of a pandemic, as you may recall, one which depended on everyone banding together to keep things from getting worse. Instead, a good half of the US decided the measures asked were too much, or that they knew better than experts.

You remember that Dr. Seuss book, "Horton Hears a Who?" I barely do, myself, but two things stand out in my memory:

First, the titular elephant is trying to save the microscopic universe of Whoville, and faces ridicule because "anything that can't be seen or heard is nonexistent." In reality, rather than a children's book, those naysayers would win, because all it would take would be one of them to mess up everything (I've referred to this as "Lone Asshole Theory," but the truth of it is, in these cases, it's not just one lone asshole, but a whole group of them).

And second, focusing instead on the microcosm of Whoville itself, it turns out (spoiler alert) that the only way to save the village is for everyone in the village, without exception, to make enough noise so that the aforementioned macroscopic deniers can hear them. Almost everyone makes noise like it's Purim and the rabbi's about to say the name Haman. Once the last holdout is convinced to scream "Yopp!" (which I'm almost certain is a Walt Whitman allusion), the disbelievers finally obtain the evidence of their own senses. Again, in reality, as opposed to a children's story, about half of Whoville would disbelieve that all the noisemaking is necessary, and a significant number of them would file complaints against their neighbors in the Mayor's office—that is, they would, if they weren't about to be utterly destroyed in an apocalyptic event.

Not only that, but as the deniers had already made up their minds, no amount of evidence would ever convince them otherwise.

Yes, people have misinterpreted that book for their anti-choice agenda, too. And those people are also part of the problem.

Apart from climate change and pandemic denial ("anything that can't be seen or heard is nonexistent"), there was one more instance, after I wrote that blog entry linked above, that reinforced my conviction that we'll never get enough people to agree on a course of action to produce meaningful change, and that was the 2022 World Cup.

Going in, we all knew, or should have known, about the host country's use of involuntary labor, housed in foul conditions and worked to the bone, to build the shiny facilities for the event. We had an opportunity, then, to speak on their behalf, to sound our not-so-barbaric Yopp over the roofs of the world. To send a clear message to all the sponsors and advertisers and organizers: "We will not stand for this." But no... we (by which I mean "you," because I did not watch) sitting at home or gathering in sports bars or even attending in person, sent a different message: "Please, go ahead and enslave people, so long as our entertainment is cheap or free."

This sort of thing is why urging individual action will never work to solve our problems, or to save the unheard, invisible people.

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