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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QPdoll

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I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

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Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

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WakeUpAndLive️~Happiness

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last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
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Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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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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