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 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🎁

about 24 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

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

December 22, 2024 at 6:24am

#1081398

Revisited: "The Trouble with Quibbles"

As usual for Sunday entries these days, I selected an older entry at random to take a second look at. This time, I didn't land all that far in the past, just over a year (one year being my minimum for these exercises), and found this: "The Trouble with Quibbles"

Being relatively recent, the linked article,

from Vox, is still up, and I didn't see any indication that it's been revised since then.

I will therefore address, primarily, my own thoughts at the time.

Quibble 1: "Intelligent." I've railed on this before, but, to summarize: What they really mean is "technology-using."

I have, in fact, bitched about this sort of thing on numerous occasions, and for reasons I go over in that entry. But, even apart from the tiresome jokes about humans not being intelligent, we know of other intelligent life on Earth: octopodes, dolphins, cats, crows, etc. It took entirely too long, from the perspective of our time as a species, to recognize these intelligences. Our communication with them is limited in scope; those species are related to us, so you'd think there would be enough common ground to establish a dialogue, but no. How much worse might it be to communicate with someone from an entirely different genetic lineage?

Of course, there's always the most basic form of communication: violence. I know it's fashionable to think that any culture advanced enough to get to the stars will have put all that behind them, but I'm skeptical. We certainly haven't. Humans fear the Other, and there's probably sound evolutionary reasons for that, but nothing would be more Other than space aliens. To them, we're the space aliens.

We're looking for signs that some extraterrestrial species has developed communication or other technology whose effects we can detect. This technology would indicate what we call intelligence, but not all intelligence develops technology. One might even successfully argue that it's kinda dumb to invent technology.

Quibble 2: UAP may be more or less silly than UFO, but I believe it to be a better fit.

UAP may have less baggage than UFO, but we have a history of taking new, neutral terms and giving them the same encrustations of connotation that we give the old terms. Like how "retarded" started out as a value-neutral term to describe humans of lower intelligence (see above), to replace words like idiot, cretin, and moron, which had turned into insults. Then "retarded" turned into an insult, and some say we shouldn't be using the term at all. Well, that's special.

Point is, I give UAP (unidentified anomalous (originally aerial) phenomena) about five more years before they have to come up with something new because UAP studies have taken a turn for the edge of the world.

And I don't doubt that there's something to study. Sure, there are hoaxes; there are always hoaxes, like with Bigfoot, but they're probably not all hoaxes. I just don't jump straight to the conclusion that if there's a sighting of something that can't be immediately identified, it must therefore be aliens. That's just retarded.

Quibble 5: What's the first thing we did when we started exploring space? Sent robots, not people. No reason to assume hypothetical aliens wouldn't do the same.

This can probably be nitpicked because some of our early ventures into space were crewed: first person in space, first human on the moon, etc. Still, Sputnik (not really a robot but not living, either) preceded Gagarin, and lunar probes preceded Tranquillity Base, and since then pretty much everything outside of Low Earth Orbit has been a robot.

Well, that's all I'm going to expand on today. My thoughts haven't changed much in the 14 months since that entry, and we have found no extraterrestrial life, intelligent or otherwise, during that time, so the search continues.

2 Like • Comment

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