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 7 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?

December 25, 2024 at 9:01am

#1081497

A Mammoth Discovery

I've banged on in here many times over what is and is not an illusion. Here's a bit from Popular Mechanics (way back in 2020) about things we know are illusions.

Why We've Fallen for Optical Illusions for Thousands of Years
Mammoth or bison? Rabbit or duck? Ambiguous images have tricked our eyes forever.

Forever, huh? Our eyes haven't been around forever.

Okay, okay, no, I'm well aware of the emphatic connotation of "forever," and it's beneath me to rag on that usage. I did it anyway.

So, because pictures in here are hard to do, you'd need to visit the link to see the oldest known optical illusion: a carved mammoth / bison dated to 14,000 years ago, thus showing that humans have been fascinated by optical illusions, if not forever, for at least 14,000 years.

Like many optical illusions, these images play on the human brain’s urge for context and turn our first impressions upside down.

I say it also plays on our predilection for pareidolia. Much of art does, really; it can be deliberately-induced pareidolia, which is why we recognize the smiley-face emoji as a smiley-face. These illusions deliberately induce pareidolia in more than one way.

The ambiguity itself is the point, and researchers have studied how the regular human experiences and preconceived notions we all carry around influence the way we decide between ambiguous options or fill in missing information.

And there's a metaphor in there, somewhere.

There's not a lot more at the article, which doesn't even acknowledge my hypothesis about pareidolia, preferring instead to talk about brain plasticity and resolution of ambiguity. But it did get me wondering why these illusions don't elicit the same kind of groaning hatred that their linguistic equivalent, the pun, does?

Maybe because evolution has worked to keep us punsters from contaminating the gene pool.

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