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🎁

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

October 17, 2024 at 10:19am

#1078457

Pros and Contrails

In a sane world, this PopSci article would put an end to one of the stupidest conspiracy "theories" immediately.

Why airplanes leave white streaks in the sky
Contrails are a problem, but not how you might think

Of course, in a sane world, the "theory" would never have gotten traction in the first place.

If you’ve ever looked up at a mostly blue sky and seen straight white lines criss-crossing the horizon, or watched a plane puff out a plume as it passed above, you may have wondered, “what causes that?”

Yeah, back when I was a kid. This was before the internet, so I used the version of Wikipedia we had back then: I asked my dad, who launched into a long-winded explanation of what he called vapor trails.

I think that was his way of keeping me from asking too many questions: answer so thoroughly that I'd get bored and stop asking questions.

And no, they’re also definitely not “chemtrails.”

I'd like to think that, even without the influence of my father's scientific mind, I still wouldn't have bought into that idiocy. But I can never be sure.

What they are, instead, is frozen water vapor crystallized on soot particles, both of which are standard byproducts of a jet’s combustion engine.

Well, there you go: the answer.

And though conspiracy theories about airplanes distributing mind control chemicals aren’t correct, contrails are having a negative planetary effect.

Because of course they are. Nothing we do is allowed to actually be good for the environment.

Atmospheric conditions have to be just right to enable the jet vapor to crystallize: moist enough that the water doesn’t evaporate, and cool enough that it freezes.

But as everyone knows, thanks to the internet, nothing that's described as "moist" can ever be cool.

A growing body of scientific research, including studies conducted by both Barrett and Stettler, indicate that contrails cumulatively have a warming effect, contributing to human-caused climate change.

These studies are important, because the leading hypothesis that I'd heard was that, being basically cirrus clouds, they'd have the effect of reflecting some sunlight, reducing the "heat in" side of the climate warming equation.

Perhaps they do, but other things about them overshadow (pun intended) that.

Since contrails, like other clouds, are bright white and insulating, they reflect light and heat, and also trap it. On a sunny day, a contrail does two things simultaneously. It “acts like a blanket,” preventing heat radiating from Earth’s surface from escaping to space, says Barrett–this is the warming effect. Simultaneously, it also reflects sunlight from space away from Earth’s surface, in a cooling effect. Unfortunately, even when the sun is shining, generally the blanket effect outweighs the reflector effect, says Barrett.

At least according to this guy.

Anyway, the article goes into what can be done about them, and, surprisingly, the answer isn't "just stop flying" or "stop using plastic straws." In other words, this is one contributor to climate change that can be mitigated fairly easily, and it's not even about us doing penance for how sinful humans are (just corporations).

And yet, some of us are just going to blithely keep on believing that they're mind-control chemicals.

1 Like • Comment

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