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 14 hours 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 ✍️

about 17 hours 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

about 20 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day 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

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

🌖 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

May 23, 2024 at 10:01am

#1071563

Across the Sea

Appropriately from Atlas Obscura, the tale of a great world explorer:

The First Viking Woman to Sail to America Was a Legendary Traveler
Back when the Icelanders called a part of Canada the “land of grapes.”

Now, it's possible that we shouldn't be using the word "Viking" like that, as I mentioned long ago here: "The Once and Future Viking"

. But I'm just quoting the article here.

Her full name, in modern Icelandic, is Guðríður víðförla Þorbjarnardóttir—Gudrid the Far-Traveled, daughter of Thorbjorn.

I'll just note here that some of those weird-to-us letters used to be in English, too.

She was born around 985 AD on the Snæfellsnes peninsula in western Iceland and died around 1050 AD at Glaumbær in northern Iceland.

Just looking at that, one might conclude that she didn't travel very far at all.

What little we know of her comes from the Saga of Erik the Red and the Saga of the Greenlanders. These are collectively known as the Vinland Sagas, as they describe the Viking exploration and attempted settlement of North America—part of which the explorers called “Vinland,” after the wild grapes that grew there.

A few entries ago, some article said that meant "Land of Wine," which may be inaccurate, but I like it better anyway.

Also, they freely mix fact with fiction. Their pages crawl with dragons, trolls, and other things supernatural.

How else are you going to scare the kiddies into behaving?

But the central tenet of the sagas has been proven by archaeology: In the 1960s, the remains of a Viking outpost were dug up at L’Anse aux Meadows, on the northern tip of Newfoundland.

I wouldn't say "proven." "Supported," maybe. It's not news anymore that Scandinavians made it to North America long before Italians did.

Among the rubble was found a spindle, used for spinning yarn, which was typical women’s work and thus possibly handled by Gudrid herself.

Right, because Gudrid sounds like the kind of chick who would do "typical women's work."

And in the Saga of the Greenlanders, Gudrid is called “a woman of striking appearance and wise as well, who knew how to behave among strangers.” That’s a trait that may have come in handy when dealing with the Native tribes of North America, whom the other Vikings dismissively called skrælings (“weaklings,” “barbarians”).

As I've noted before, who's the "barbarian" depends on who you're asking.

The article continues with a summary of the sagas involving Gudrid, and while I'm sure the originals (well, the original written-down versions, I mean) would be fascinating, the Cliff's Notes here seem to provide the pertinent details.

Another story from the sagas that has mystified readers for centuries because it mentions two “Gudrids” and has traditionally been dismissed as a ghost story could in fact be the earliest recorded conversation between a European and an American.

And no, they didn't discuss trade agreements or war. Or much of anything, considering they apparently didn't have time to learn each other's languages.

There's a lot more at the article, but as it notes, her relatives Erik and Leif got all the PR, but they didn't travel alone.

3 Like • Comment

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