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 20 hours ago
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🎁

about 24 hours ago
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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day ago
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?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 26, 2024 at 11:04am

#1080516

Inevitable Two

This one from Mental Floss reminds us of our mortality. The article is a couple of years old, but I'm sure the subjects remain deceased. I did a similar entry a few years ago: "Inevitable"

. And I Revisited that one just a few weeks ago: "Revisied: "Inevitable""

13 Authors Whose Deaths Were Stranger Than Fiction

Authors are used to killing their darlings and that sometimes means offing characters in creatively unconventional ways.

Non-authors, too, die on a predictably regular basis. But when they die in a weird way, it's usually not irony.

1. Sherwood Anderson

As far as I know, I've never read anything by this author, so now the manner of his death is the only thing I know about him.

An autopsy revealed the culprit to be a 3-inch-long wooden toothpick in an olive that the author had swallowed while enjoying a martini.

Having imbibed more than my share of martinis (which, incidentally, have to contain gin and vermouth and an olive garnish, otherwise it's not a martini), I can say with some certainty that one would need to imbibe a whole hell of a lot of them to not notice a 3-inch toothpick entering one's digestive system. So many, in fact, that it's not the toothpick that I'd be worried about.

2. Aeschylus

Oh, we're going old-school now. This one was in the earlier entry, though at the time I apparently accepted the story at face value.

According to writer Valerius Maximus, Aeschylus was hit by a falling tortoise while sitting outside Sicily’s city walls: “An eagle carrying a tortoise was above him. Deceived by the gleam of his hairless skull, it dashed the tortoise against it, as though it were a stone, in order to feed on the flesh of the broken animal.”

The truth may be stranger than fiction, but that particular story has all the hallmarks of being, well... fiction.

3. Gustav Kobbé

Music critic and author Gustav Kobbé loved to sail, but the hobby led to his death.

Whereas with most music critics, pissing off musicians is what does them in.

4. Margaret Wise Brown

In 1952, Goodnight Moon (1947) author Margaret Wise Brown was in France on a publicity tour when she developed appendicitis and was taken to the hospital for emergency surgery.

As the article notes, it wasn't the appendicitis that killed her. Not directly, anyway.

5. Tennessee Williams

Dr. Annette J. Saddik, Distinguished Professor of Theatre and Literature at the City University of New York, explained in 2010 that the false cause of death was due to John Uecker, Williams’s assistant, telling “the Medical Examiner, ‘Look, people are going to think it’s suicide or AIDS or something bizarre and we don't know what happened.’ So the Medical Examiner, said, ‘OK, he choked on a bottle cap.’”

A rare case of fiction being stranger than truth. In any case, I think the last article of this sort that I featured also brought up Williams, but apparently, the "bottle cap" story was taken as true, there.

6. Pietro Aretino

Pietro Aretino was an Italian satirist, playwright, and poet, who is credited with inventing written pornography.

Say what, now? It's a near-certainty that as soon as writing was invented, someone used it to make porn. In any case, as with the ancient Greek above (and some of the others here), the story of his death is questionable.

7. Sir Thomas Urquhart

The Scottish writer and translator died in 1660, supposedly because the news that Charles II... had retaken the throne caused him to burst into a fit of joyful, but deadly, giggles.

Look, if you're going to claim these are true death stories, at least make them true death stories.

8. Edgar Allan Poe

I want to say "we all know this one," but there's always someone learning something for the first time. In this case, you'll just have to check out the link. Or read the earlier entry. Or look up his Wiki page. The circumstances surrounding his death were mysterious, but well-reported.

9. Sir Fulke Greville

Dying on the toilet isn’t the most dignified way to go, and though Elizabethan poet and dramatist Sir Fulke Greville managed to avoid that fate, the toilet certainly played a part in his death.

Eh, that's a stretch.

10. Mark Twain

Another really famous one, and as far as I know, well-documented. I remember as a kid being told about this, and thinking, "What a shame that he never got to see the comet," though he might have seen it before he died; I don't know. In any case, I couldn't see it when it swung through in 1986, so I know I'll never see it.

11. Molière

Legend often has it that Molière died onstage, but that’s not actually true.

Yeah, and I wonder about some of these others.

Some of Molière’s lines as Argan were eerily prophetic of his imminent demise.

Look, if you talk about death, and then die, it's not all that prophetic. Everyone dies, and lots of people talk about it beforehand.

12. Dan Andersson

On September 16, 1920, Swedish author and poet Dan Andersson checked in to the Hotel Hellman in Stockholm, settling in Room 11.

And this is why we don't have "Hell" in the name of hotels today.

13. Francis Bacon

The only ironic death I would have accepted here would be "fried on a stove." The actual story, if true, does involve food, though. And science.

There's a deep human curiosity to know how someone died. I can understand that; most of us want to know if it's something we should avoid.

Failing to avoid a particular method of demise, however, should definitely warrant internet fame.

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