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 11 hours ago
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

a day ago
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

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

July 2, 2024 at 9:00am

#1073461

Imma Let You Finish, But...

The current round of "Journalistic Intentions"

[18+] features prompts relating to the Olympics. This is a subject I know almost nothing about, so I get to stretch my superpower of being able to write about any topic.

To make it even more potentially difficult for myself, I'll be picking the prompts (there will be eight of them, including this one) at random from the list.

Vanderlei de Lima

I've long thought of the Olympics as the exclusive province of young people. Well, in the sports, anyway; I imagine the IOC as a bunch of septuagenarians proclaiming this and that from on high (I don't know who actually composes that body).

And by "young," I generally mean "teens and early twenties." Then you retire, do commercials, and eventually get involved with a family who's famous for being famous. Hell if I know; it's so hard to Keep Up these days.

But, according to this guy's Wiki page,

dude was 35 when he ran in the 2004 games.

For context, I'm only three years older than him. For more context, Springsteen was 35 when he released Born in the USA (but much younger when he released Born to Run, which might have been more appropriate for Lima).

From that Wiki page:

Vanderlei Cordeiro de Lima (born 4 July 1969) is a Brazilian retired long-distance runner.

Now, see, I'd have guessed Peruvian from the name. Shows how much I know. (And no, I've never even heard his name before, to the best of my recollection.)

While leading the marathon after 35 km at the 2004 Summer Olympics, he was attacked on the course by Irish former priest Cornelius "Neil" Horan.

There's gotta be a story there. Fortunately, the article later expands upon it:

Soon after the 35 km (22 mi) mark, holding a lead of around 25 seconds, Lima was halted and grappled with by spectator Neil Horan, an Irish priest who was later defrocked.

The implication is that Horan was actively serving as a priest at the time, so saying he was attacked by a "former priest" up there is a bit misleading.

What Godly or worldly reason could Horan have had for being a human stumbling-block?

Horan said "I wasn’t doing it as a prank, I was doing it to spread the gospel and to prepare people for the second coming.”

Religious "logic," folks. I know if I wanted to convince someone my invisible sky-wizard was about to press the big red reset button, I couldn't think of a more appropriate way to spread the message than to fuck up someone's marathon run.

2004 was before social media infected the internet, but the internet itself was absolutely a thing then. I know this because that's the year I joined WDC, not long after the Olympics.

And here we are, 20 years later, and despite every effort and fervent prayers, the second ain't come yet.

In any case, there's not much else at the Wiki link besides Lima's other accomplishments in putting one foot in front of the other faster than everybody else. He did win sportsmanship awards, though, so I figure that counts for something regarding his character.

Which is more than I can say for his attacker.

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