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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 9 hours 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

a day 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🎁

a day 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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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

September 27, 2024 at 11:05am

#1077349

Bibo Ergo Sum

Mystery solved! (Maybe.) According to the guide yesterday, at least, the city of Tours is named after the Celtic tribe that used to live there. I guess it's kind of like America where we ran off or killed all the natives then named some towns and rivers after them. But it turns out that the city's coat of arms features three black towers. Yes, that's right, the city's heraldry is a pun.

That sort of thing isn't unheard of. The only other one I can think of off the top of my head, though, is the Northern England town of Berwick, whose coat of arms features... a bear.

This wasn't the only thing I learned yesterday, during my guided tour of some of the wine country east of Tours. Another cave was involved, this one belonging to a winery in Vouvray. Vouvray being some of my favorite wine, this particular tour was the primary reason for my trip, and it did not disappoint.

Part of the reason Vouvray tastes like ambrosia of the gods, while other wines made with chenin blanc grapes in other areas of the world taste like piss, is that there's a unique combination of soil, climate, and process (among other factors) in this tiny part of France. What I learned was that part of the terroir is because they force the vines to dig their roots deep, very deep (the guide said something like 10 meters is possible) into the ground, where they delve into limestone.

Limestone is a fossil rock. Billions of years ago, tiny sea creatures died and their skeletons built up and got compressed and turned into limestone over the eons. So the flavor of my favorite wine owes its greatness to the death of these long-ago organisms, thus demonstrating once again the circle of life.

And also, it seems Tours has a thing for Rene Descartes, the scientist/mathematician/philosopher I'm sure I've mentioned in here hundreds of times. One example is here: "I Drink Therefore I Am"

I understand he was born in Tours, or at least somewhere nearby. My hotel was, in fact, right next to a statue of him, inscribed with his most famous quote.

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