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 6 hours 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.

1 Like

🐕GeminiGem🎁

about 10 hours 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

about 12 hours ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

about 12 hours 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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2 Like

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

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 4, 2024 at 8:17am

#1079484

It's Hard Not To Scare

This Cracked article might have been better to post before Halloween, but remember, tomorrow is Election Day here. Scary stuff is still in our future.

Scientific Studies That Sound Like the Beginning of a Horror Movie
Two words: Spider goats

I have often looked at articles about what some scientists are doing and think: "You fools! Have you never seen a horror movie?"

Animal testing is an unfortunate but necessary part of certain scientific fields — that is, until the scientists decide to go all Dr. Moreau on some rats. Then it’s just weird. It’s like they’ve never seen a single monster movie.

See? I'm not the only one. (Sometimes, "horror" gets replaced with "science fiction," but usually, horror is still a subgenre.)

5 The Spider Goat

Spider silk is super useful for a lot of different things, but there’s a reason you don’t see a lot of spider farms.

I saw a video recently on how silkworms do their thing. Silkworms are larval moths, and generally unpleasant to look at, but they lack the visceral horror of arachnids. Still, spiders are cool... from a distance.

In 2012, Utah State University geneticists rectified that problem by splicing spider DNA into goat embryos, who eventually grew up to lactate spider silk.

Friendly neighborhood spider-goat.

4 The Man Mice

In 2013, scientists at the University of Rochester implanted human glial brain cells into the brains of newborn mice, who became much smarter and learned faster than other mice as a result.

"What are we going to do tomorrow night, Brain?"

Also, anyone who has ever seen a horror movie should have implored them to stop.

3 Acid Elephant

In the ‘60s, experimenting with LSD was all the rage, in both the scientific and “hanging out in your friend’s cousin’s basement” senses of the word. By 1962, scientists at the University of Oklahoma had run out of ideas until one of them asked, “What if we gave an elephant a thousand doses?”

Which makes me wonder: if we see pink elephants, what do elephants see? Gray humans?

2 Magnetized Cockroaches

“Can you turn a cockroach into a magnet?” is a question you only ask after you’ve been seriously scientifically jaded...

...or on a serious acid trip.

1 Zombie Dogs

Believe it or not, we do know how to bring once-living creatures back from the dead.

Thus shifting the definition of "dead." Used to be: heart stopped = death, but then CPR came along, and now doctors have to pretty much guess when the point of no return occurs.

Cornish hoped to try his method on humans, specifically a recently executed prisoner, but the government forbade it not out of any fear of a zombie apocalypse but because they weren’t sure how double jeopardy laws applied to a revived corpse.

Seems like an important legal loophole to fix.

Add to this mix the penchant of certain scientists to revive millennia-old bacteria found in ice cores, and you definitely have the makings of a horror movie. But still not as scary as tomorrow's election.

4 Like • Comment • View 3 comments

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