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

December 20, 2024 at 10:25am

#1081341

To Err Is Human

Solstice tomorrow (around 4:20 am here), so this is my last entry of astronomical fall. Today's article, from BBC, has nothing to do with seasons, though, and it's a subject I really shouldn't be weighing in on—but of course, I'm going to do it anyway.

Humans may not have survived without Neanderthals

What inspired me to jump in above my head here was the lede:

Far from triumphantly breezing out of Africa, modern humans went extinct many times before going on to populate the world, new studies have revealed.

Now, there's a poorly phrased sentence if I've ever seen one. It should be blindingly obvious to everyone who can read this that modern humans didn't go extinct. This is a fact on par with "Earth is roughly spherical" and "Space is mostly vacuum." Actually, wait, no, I'm even more sure that modern humans didn't go extinct than I am about those other things, because, last I checked, there were about 8 billion modern humans running around. Or sitting around. Whatever. You do you. Point is, we're not extinct yet.

Likely, the author meant "sub-populations of modern humans went extinct many times," which, okay, I guess they have science to back them up on that, and I'm not going to argue about it. But I feel like the way it's phrased would be like if they said "humans went extinct in Pompeii in 79 C.E."

The new DNA research has also shed new light on the role our Neanderthal cousins played in our success.

This is, I think, the interesting bit here. But I'd like to emphasize the "cousins" metaphor there. Sapiens and neandertals (the spelling of the latter appears to have legitimate variants) shared a common ancestor. A certain ape population separated at some point, genetic drift and selection occurred differently in each population, until you got groups with clear physiological differences in the fossil record. But, apparently, the physiological differences weren't enough to prevent interbreeding.

The definition of a species is, to my understanding, a bit of a slippery concept in biology. That is, it's not always obvious what constitutes a separate species. If it were as easy as "a population that can breed to produce fertile offspring," we wouldn't consider sapiens and neandertals separate species because, according to DNA evidence, they produced fertile offspring together.

While these early European humans were long seen as a species which we successfully dominated after leaving Africa, new studies show that only humans who interbred with Neanderthals went on to thrive, while other bloodlines died out.

Again, I feel like this is poorly phrased, and puts too much emphasis on Europe. Apparently, there are populations in sub-Saharan Africa today with no neandertal genes and, again, obviously they didn't die out. And they're the same species as the rest of us mostly-hairless bipeds.

Apart from these nitpicks, I think the new findings are fascinating, delving into how, basically, hybridization led to greater hardiness. As with all science, it may be overturned or refined through later studies, but this article itself describes an overturning of previous hypotheses about early human ancestry. And it has helpful infographics and pictures.

But unless we invent time travel (unlikely), all we can do is make hypotheses and test them. It's really a very human thing to do.

4 Like • Comment • View 3 comments

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