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 4, 2024 at 9:38am

#1073524

He Said, Archly

Another random prompt from "Journalistic Intentions"

[18+] today.

St. Louis 1904

There are, in the world, very few cities with icons more iconic than the Gateway Arch. There's the Eiffel Tower, of course, arguably the most recognizable structure in the world. Across the Channel, there's the unmistakable steampunk intricacy of Elizabeth Tower, known worldwide but usually mistakenly labeled as Big Ben (which may be a case of metonomy, as Big Ben is the largest bell within the tower, but is more likely just mistakery). You have the sweeping, nested shells of the Sydney Opera House, the Washington Monument, maybe the Space Needle, CN Tower, and the Empire State Building. I'd be remiss if I didn't mention the Pyramids, but they're not technically a city symbol, located as they are outside the boundaries of Cairo.

Almost all of these establishment-shot-bait constructions have one thing in common: the "tower" bit (the Opera House being an exception). This inevitably leads to jokes about phalluses (Phalla? Phalli? Hell if I remember what Latin declension it belongs to), especially in the painfully obvious case of the Washington Monument (Father of his Country and all that).

Not so with the Gateway Arch. It's not a tower; it's... well... an arch. Two foundations, not one. Curvy, not straight (unless you view it from one side or the other, but who does that?) I'm not saying it's the equivalent expression of feminine potency, mind you; just that it's not an erection sticking straight up into the sky.

Thus, it might surprise younger folks and those not familiar with the US in general, the State of Misery, and St. Louis in particular, that the Arch is barely 60 years old.

Thus, when one considers the condition of St. Louis in 1904, one must utterly wipe any image of the Gateway Arch from one's internal visualization of that important Midwestern city.

That year, St. Louis, just over 100 years after the Louisiana Purchase, hosted a World's Fair, formally known as the Louisiana Purchase Exposition.

For other context, consider these historical milestones:

The Wright Brothers' first flight wasn't even a year in the past, having taken place in the middle of December 1903—but the Exposition hosted the first-ever airship competition (see above Wiki link). This led to St. Louis being just as associated with flight as Kitty Hawk, a legacy later immortalized in the name of Lindbergh's famous transatlantic airplane, the Spirit of St. Louis. (Okay, the plane's name had nothing to do

with StL's history regarding flight, and these days, you're taking your life, or at least your sanity, in your hands by flying Spirit across the Atlantic.)

Ford had yet to democratize the automobile, and the hot topic in cities around this time was streetcars. I have no desire (pun absolutely intended) to get into the arguments concerning these modes in this entry, and anyway, San Francisco became more associated with trolleys. This also led to more than a century of pain for philosophy students as they contemplated various permutations of the Trolley Problem.

And then, of course, there were the Olympics.

Apparently, the 1904 Summer Olympics

were meant to be held a bit further north, in Chicago, the first such event to be hosted outside Europe. In typical American fashion, the Expo organizers apparently stomped their feet and shouted, "NO!" So the games were moved to Misery, which, in those days, was kinda hard to get to (remember, airplanes were less than a year old; overland routes sucked, and navigating up the Mississippi takes time).

But, regardless, it turned world attention to the site that used to be a major Native American gathering place (see my April entry, "Things Go Better With Cahokia"

, where, apparently, a lot of people came from all over to participate in social and sporting events.

Some things, it seems, are universal.

At least the Gateway Arch is unique.

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