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 1, 2024 at 8:39am

#1065309

They Ain't Here

The truth is out there... but you don't believe it. From New York, not to be confused with The New Yorker:

No, Aliens Haven’t Visited the Earth
Why are so many smart people insisting otherwise?

Just to get this out of the way: Smart people can still draw bogus conclusions. They can be fooled. They can be exceptionally good at fooling. I wouldn't give a smart person's opinion more weight, unless it's in a field they're credentialed in, any more than I'd trust a rich person's opinion over that of a poor person.

Last month, Sean Kirkpatrick, the head of the Pentagon office responsible for investigating unexplained aerial events, stepped down.

The article's from the end of January, so I guess that happened in December?

He said he was tired of being harassed and accused of hiding evidence, and he lamented an erosion in “our capacity for rational, evidence-based critical thinking.”

I feel that.

He may have been pushed over the edge by a pair of events from the past summer. In June of last year, Avi Loeb, an astronomer at Harvard, announced that he had found some tiny blobs of metal by dragging a magnetic sled over the bottom of the Pacific near Papua New Guinea. He claimed that these blobs were metallic droplets that had melted off an interstellar object that might have been “a technological gadget with artificial intelligence” — the product of beings from another star system.

That's the same numbskull who claimed that the interstellar wanderer called 'Oumuamua

was a product of technology.

Look, most of the time, I see Avi Loeb's name, Harvard gets mentioned. I think that's an attempt to give him some credentials. But it has the opposite effect on me: instead of being more inclined to believe Loeb's crackpot "it-must-be-smart-aliens" conclusions, I become more inclined to dismiss anything that comes out of Harvard.

In other words, that guy's such a disgrace to that prestigious university that his association with it (along with a few other Harvard-associated people recently) decreases its prestige.

Oh, and sure, random blobs of metal could have come from an interstellar visitor. I don't consider the odds of that to be very high. Claim like that, though, you're going to have to rule out terrestrial origins, first.

Thoughtful, sensible-seeming, non-crankish people at Harvard, at The New Yorker, at the New York Times, and at the Pentagon seemed to be drifting ever closer to the conclusion that alien spaceships had visited Earth.

One of those things is not like the others...

To be clear, we should investigate these claims, within reasonable budgetary limits. Though any debunking that gets done sure won't convince the UFO nuts. On the other hand, show me definitive evidence, and I'll be convinced.

Yet even after more than 70 years of claimed sightings, there was simply no good evidence. In an age of ubiquitous cameras and fancy scopes, there was no footage that wasn’t blurry and jumpy and taken from far away. There was just this guy Grusch telling the world that the government had a “crash-retrieval and reverse-engineering program” for flying saucers that was totally supersecret and that only people in the program knew about the program. Grusch said he had learned about it while serving on a UAP task force at the Pentagon. He interviewed more than 40 people, and they told him wild things. He said he couldn’t reveal the names of the people he interviewed. He shared no firsthand information and showed no photos.

And that, folks, isn't evidence. It's crackpottery, albeit perpetrated by someone who would be in a position to know. It makes me wonder what he was paid to distract us from. Yes, I have my own conspiracy leanings, but they involve human activity, not purported alien.

Now, the article goes on for a while, and it's absolutely worth a look, at least a skim. I won't comment on it further, though, except to say that our collective fascination with the subject says nothing about aliens and everything about humans.

2 Like • Comment

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