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:

tj-Merry Mischief Maker

about 7 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

about 12 hours 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

about 14 hours 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 ✍️

a day ago
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

a day ago
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

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

April 21, 2024 at 9:17am

#1069306

Revisited: "Hack This"

I'll be on what passes for vacation for me this week—I have nothing to vacate from, but I do like to go elsewhere from time to time—so entries will happen whenever I get a chance and probably short and even more pointless than usual.

For today, though, I'll do my usual Sunday thing and look back at an old entry. This one's from 2020: "Hack This"

Worth reading if you want a decent takedown of "lifehacks." Or even if you don't. Especially if you don't.

I still see "lifehacks" from time to time, but I see more parodies of them, which warms my heart. Things like: "Life hack: Don't have a mental breakdown at home. Have it at work so at least you're getting paid for it."

The article I linked

(Medium, 2017) is still up as of today. And my opinions haven't changed much, but maybe a little, and maybe some points need clarification. But first, I'll address the end of the entry, where I discover, too late, that this advice article that is a takedown of advice books and articles is actually an ad for the author's advice books.

Also, I hate reading this far along a halfway decent article only to find that it's a commercial in disguise. Bah.

Yes, I have said numerous times that I don't mind taking a look at book ads here, on a site that caters to readers and writers, so long as the content is worth commenting on. This one, however, did manage to catch me by surprise. Usually the book-flogging is near the beginning, or in a sidebar, or otherwise obvious when you start reading the article. This one was, I felt, deceptive—moreso because, as I wrote then: "I feel like he has good points, but those are somewhat muted by the fact that he's doing exactly what he's railing against."

Now... one might say, "But Waltz, here you are doing it too." Yes. I am. But I'm not trying to sell anything.

Besides, if I really objected to it, I'd have scrapped the entry and done a different one.

Rereading this entry, I realize it might read as if I'm against self-help books and articles in general. I'm not, necessarily. It's just that most of them only "self-help" the author to make money. There's nothing wrong with making money; most of us want to do that. But doing it by misleading others into doing something that doesn't help, and may even actually harm, is generally called "fraud," and is frowned upon.

So, just one more comment on my previous comment:

I keep seeing that the true enemy is "processed" or "overprocessed" foods, but I haven't found a good definition for those, yet. I mean, technically, cooking is a process, and - raw-food-diet bullshit aside - cooking is what makes a lot of food more nutritious and digestible. Potatoes, for example. It's probably what allowed our ancestors to evolve these great big brains that most of us don't use.

Obviously, it's been four years, so a) "processing" has been better-defined, while at the same time I haven't seen much about it lately; I wonder what the next anti-fad will be. And b) I shouldn't have typed that last sentence; it's misleading and does what I'm railing against here: assertion without evidence or experience. Not that I'm above doing that, but in this particular case, it's evolutionary guesswork, which I have issues with, and it's also what's commonly known as a "chicken/egg" scenario: did we evolve big brains because our ancestors cooked their food, or did they cook their food because they had big brains from some other adaptation? Or was it a synergy of some kind? My only point should have been that cooking food is good.

In reality, the "chicken/egg" scenario is easily resolved: eggs existed long before what we call chickens, and the first chickens hatched from eggs laid by some dinosaur descendant that was almost a chicken. That's still evolutionary speculation, but at least in this case, it fits with what we know of evolution.

Therefore, it was the egg that came first.

Fortunately for chickens and those of us who enjoy eating them, it wasn't cooked.

3 Like • Comment • View 7 comments

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