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 20 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.

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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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 27, 2024 at 8:41am

#1080556

Son of an Itch

I'll admit it: I'm only linking this NPR article because the headline is glorious.

Chronic itch is miserable. Scientists are just scratching the surface

I mean, I come up with good puns sometimes, sure. But that one nailed it. (Nail? Scratch?... No? Okay.)

We've all had bug bites, or dry scalp, or a sunburn that causes itch. But what if you felt itchy all the time — and there was no relief?

Well, then you know you've truly sinned in the eyes of the Lord.

Seriously, though, that happened to me. Not, like, permanent, the way these unforgivable souls suffer, but a side-effect of some medication I had to take for a few weeks.

Journalist Annie Lowrey suffers from primary biliary cholangitis (PBC), a degenerative liver disease in which the body mistakenly attacks cells lining the bile ducts, causing them to inflame. The result is a severe itch that doesn't respond to antihistamines or steroids.

Yeah, that's gotta suck.

PBC is impacts approximately 80,000 people in the U.S., the majority of whom are women. At its worst, Lowrey says, the itch caused her to dig holes in her skin and scalp. She's even fantasized about having limbs amputated to escape the itch.

"PBC is impacts?" NPR, you used to be better than this.

Also, while I totally get the limb-amputation fantasy, seems to me that would just make the itching worse, what with phantom limb syndrome and all.

And finally, 80 thousand? I know that's not much on a percentage basis, but my whole city doesn't have 80,000 people in it.

There's a bit more at the link, mostly going into some detail about the phenomenon of itch. Some of it is accepted science, and some of it isn't (the evo-psych bits, I mean). I managed to read the whole thing without experiencing the sympathetic itch the article talks about, which is more than I can say for when there's an article about yawning.

But mostly, I linked it for the absolutely stellar headline.

2 Like • Comment • View 2 comments

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