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:

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 22 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 21, 2024 at 10:06am

#1066678

Socra-Tease

Here's one from The New Yorker that, when I first read it, I thought: This has comedic potential.

When Philosophers Become Therapists
The philosophical-counselling movement aims to apply heady, logical insights to daily life.

I mean, think about it.

"Doc, I'm distressed because I can't find meaning in life."

"That's because life has no meaning, so stop looking for it."

"Wow! That's deep! Thank you."

"Sure. That'll be $180."

"Here's $200."

"Thanks!"

"Can I have change?"

"You already did."

Around five years ago, David—a pseudonym—realized that he was fighting with his girlfriend all the time. On their first date, he had told her that he hoped to have sex with a thousand women before he died.

All the TNY pieces I've seen start out with anecdotes and then, maybe, bit by halting bit, go around and around in circles until they get to something more general. Sometimes they never even get there, which I call "The New Yorker School of Not Getting to the Fucking Point." This one contains a whole lot of anecdotes. My point, which you'll note I'm getting to very quickly, is that I'm going to skip a lot of it here. In brief, he worked with a philosophical therapist named Lydia Amir.

I might have missed it, but I don't think the article notes whether he had sex with her or not.

Between meetings, Amir sometimes gave him reading assignments: Schopenhauer, Nietzsche, Hume.

Oh yes, definitely, if I'm having an existential crisis, those are absolutely the philosophers I want to read.

Amir is one of a small but growing number of philosophers who provide some form of individual counselling. In the United States, two professional associations for philosophical counsellors, the National Philosophical Counseling Association (N.P.C.A.) and the American Philosophical Practitioners Association (A.P.P.A.), list dozens of philosophers who can help you with your problems.

Well. Philosophy tends to pay jack shit. Therapy, however, can bring in the bucks.

Philosophy is both a natural and a strange resource for helping people resolve the problems of life. Ancient philosophical traditions such as Stoicism and Buddhism focussed on practical ethics and techniques for alleviating suffering, but much modern philosophy seems to aim to express suffering, rather than reduce it.

I wouldn't call the aim of those philosophies to be "alleviating suffering," but there's always the possibility that I don't really understand Stoicism or Buddhism.

Some think that the practice of philosophical counselling should be more standardized. Others worry that philosopher-counsellors will miss serious mental-health issues. The two major American professional organizations stress that philosophical counselling can’t address certain severe psychiatric disorders, and urge counsellors to refer clients to mental-health providers when their issues do not fit a philosophical scope of practice.

Well, that's somewhat relieving. It's one thing to seek someone else's point of view, or be able to talk things through with a neutral party. It's another to have legitimate mental health issues that need more medical treatment.

So, anecdotes (and jokes) aside, this kind of thing obviously exists (in the same sense that anything can be said to truly exist), and it mildly amuses me. Does it work? All the TNY anecdotes in the world won't convince me one way or the other; I want to see a study.

I'm sure a philosophical therapist would have something to say about my need for evidence-based science. But, you know. Whatever.

5 Like • Comment • View 6 comments

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