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:

Wordsmitty ✍️

about 2 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?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

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

1 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

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

June 1, 2024 at 9:13am

#1071970

Border Dispute

Wars have been fought over less than this.

The Man Who Went to War With Canada
For centuries, the United States and Canada’s only remaining land border dispute has been kept alive by a single family.

Article is from Atlas Obscura and 2019, but the Wikipedia article

suggests that the issue hasn't been resolved over the last five years.

After probably a few hours at sea, they reached the island’s rocky shore, and managed to land among the island’s main residents: puffins, razorbills, murres, and Arctic terns.

If I were that guy, I'd have pointed at one of the latter birds and proclaimed: "It's my tern now!"

Depending on whom you ask, Machias Seal Island is either off the coast of Maine or of Grand Manan. It’s also either American or Canadian. It is the only place with this particular unsettled identity that you can actually stand on top of. Although the ownership of some stretches of water is still contested, this island—and neighboring North Rock, which is even smaller and barer—are the last crumbs of their land the two countries don’t agree on.

Just wait a few years, and climate change will settle the debate for us.

I feel obliged to point out that while I've never been to this island, it sits within easy distance of the easternmost point of the continental US: Quoddy Head Lighthouse. I visited that spit of land when I decided to travel from it to the westernmost point, in the state of Washington. Quoddy Head was the easier of the two to reach, requiring significantly less hiking and way fewer bears.

The article delves into the source of the land dispute, then:

If the world were more just, this would all be moot: The people of the Passamaquoddy Nation likely used the island long before anyone else even knew it existed. (“Machias,” also the name of precipitous local river, is a Passamaquoddy word that means “bad little falls.”) Instead, even as their identifications and affiliations have shifted, the neighbors have kept squabbling over it, like a pair of growing siblings in a shared bedroom.

Look, let's hope it stays on the "siblings" level of quarrel. We don't want to go to war with Canada. We have a history of losing said wars.

The rest of the (rather long) article delves into the American side of things, focusing, as the headline implies (look at that, a headline matching up with the text), on one person's, or more properly one family's, quest to keep the US claim to the island.

I won't quote further, but it's a way more interesting read than the Wiki page.

2 Like • Comment

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