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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Show Recent Comments

10 Most Recent Blog Comments:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 23 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

March 2, 2024 at 6:33am

#1065370

Thought Bubble

Hope you're not prone to paranoia.

Who Controls Your Thoughts?
Our minds are being coerced in covert ways.

After yesterday's article, clearly, the answer to the headline question is "space aliens."

In 2017, Simon McCarthy-Jones wrote an article about schizophrenia for The Conversation. The piece, he jokes, got read by more than two people, which, as an academic—he’s an associate professor of clinical psychology at Trinity College Dublin—was a thrill.

Except that, when I read this lede, I thought he was making a schizophrenia joke, like the old "I'm schizophrenic and so am I" one, which perpetuates the conflation of schizophrenia with MPD. Which I might be able to get away with, but not a psych professor. Anyway, no, that wasn't the intent. As far as I can tell.

Shortly thereafter, however, he found himself “just gripped by the iron claws of Facebook,” looking over and over again to see who had liked his article, who had commented on it.

Isn't there a psych disorder where you attribute your actions to some outside force, instead of taking responsibility for them? I don't mean that in a "we have no free will" way, but in a "the Devil made me do it" way.

Was his thinking being covertly, coercively controlled by external forces (in this case, a big tech company)? The experience got him wondering just what “free thought” actually was. And so he started wading into the murky waters of the psychological, philosophical, cultural, and legal assumptions about what constitutes thought—and how it could remain truly free.

His intellectual quest has exited his head, as much thought eventually does, and now exists in the form of a new book: Freethinking: Protecting Freedom of Thought Amidst the New Battle for the Mind.

Of course it's a book ad. Everything on the internet is an ad, or it's behind a paywall. Well, except for this blog, of course.

In any case, the rest of the ad is an interview.

We might want to say: No, we’re independent, autonomous thinkers. But I think we have to recognize that in front of a persuasive AI, we are in deep trouble.

In my experience, the people who shout "I think for myself!" the loudest are the most likely to follow the herd.

The first was that the right to free thought is an absolute right, based on the United Nations’ Universal Declaration of Human Rights. In the United States, it’s as close to an absolute right as there is in the Constitution. And that’s quite exciting because it means that nobody can interfere with your freedom of thought. There are instances where you can limit someone’s speech if it’s defamatory or false advertising or fighting words. But thought is unimpeachable, you can create absolute protection for people’s minds.

Well, okay, except that it's (at least so far) easy to say "you have an absolute right to free thought" when there is no possible way to read minds. Introduce a mind-reading device, though, and watch how quickly that right disappears. Sure, we can sometimes take an educated guess as to what's in someone's mind. Like if you're a shoe clerk, and some guy is in there looking at the Nike selection, you'd probably be right in assuming the guy's thinking "I might be interested in buying athletic shoes." But only "probably." After all, he might be idly gazing at footwear while wondering how he's going to trick some lady into bed that night.

There’s a quote by Daniel Kahneman, the Nobel laureate, who said: “Human beings are to independent thinking as cats are to swimming. They can do it, but they prefer not to.”

Honestly, I mostly saved this article so I could share that quote.

Astute readers may note that a lot of this is in opposition to my assertion that we don't really have free will. And it is. But lack of free will doesn't necessarily imply that it's other people pulling on our strings. Sometimes it's just the universe conspiring to keep us complacent.

3 Like • Comment • View 2 comments

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