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 14, 2024 at 9:54am

#1066257

Current Events

Remember Rebecca Black's squirmy earworm song, "Friday?" No, I will not link it. Well, this morning, I was tempted to do a parody in honor of today called "Pi Day," but my cats talked me out of it, so we all get to keep our sanity.

Unlike this guy, apparently. From Big Think:

Tesla’s pigeon: How the great inventor fell for a bird
"She understood me and I understood her. I loved that pigeon.”

The main trouble with Nikola Tesla's legacy is that his name was appropriated by a mediocre band, and then run into the ground by one of the most successful con artists in history. The only way to reclaim what shouldn't be a laughingstock, I think, is to remember the inventor's life, accomplishments, and, yes, even his apparent madness.

On a February morning in 1935, a disoriented homing pigeon flew into the open window of an unoccupied room at the Hotel New Yorker.

That hotel, though it's been through a few changes over the decades, still exists. I've stayed there. I don't think the windows open anymore; with air conditioning, it's not necessary, and maybe it prevents some jumpers.

While management debated what to do, a maid rushed to the 33rd floor and knocked at the door of the hotel’s most infamous denizen: Nikola Tesla.

When I was there, I even made a pilgrimage to the 33rd floor.

“Dr. Tesla … dropped work on a new electrical project, lest his charge require some little attention,” reported The New York Times.

"Charge?" They just couldn't resist, could they? Revolting how some people just plug in the most obvious puns.

Nikola Tesla—the Serbian-American scientist famous for designing the alternating current motor and the Tesla coil—had, for years, regularly been spotted skulking through the nighttime streets of midtown Manhattan, feeding the birds at all hours.

He invented way more than that. Some say, rather poetically, that he invented the 20th century. While a bit hyperbolic, it's not that far off the mark.

In the dark, he’d sound a low whistle, and from the gloom, hordes of pigeons would flock to the old man, perching on his outstretched arms.

Look, all I'm saying is, that's a remarkable image and if someone hasn't painted that, someone definitely should.

Tesla said that he and his bird could speak to one another mind to mind, and that sometimes, as they silently conversed, beams of light would shoot from her eyes.

This is the sort of thing I meant by "madness" above. But is it really? Or was he operating on a different level of reality? With genius like Tesla's, there's always that seed of doubt: maybe he was right, and it's the rest of us who are blind.

Tesla’s love of pigeons was an obsession with a capital O. Likely followed by a capital C and a capital D. He seems to have suffered from obsessive compulsive disorder, and his case was severe.

Anyone with only a passing knowledge of Tesla's life, and of pop psychology, would immediately jump to the same conclusion. While it can be dangerous and rude to diagnose someone from some distance in space and time, all the signs were absolutely there. I'd even throw in the possibility of autism.

I won't quote more from the article, which is fairly long, but, and I can't emphasize this enough, beautifully written. It weaves quite the tapestry of history and science, and, of course, there's a bit of Mark Twain in there.

This is where I'd usually relate the subject of the article to Pi Day, but all I could think of was the relationship between pi and the sine waves of electrical current, but maybe that's a bit too obvious while at the same time too esoteric. So I'll just leave it at that and go find me some pie.

6 Like • Comment

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