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 11 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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?

July 27, 2024 at 8:33am

#1074446

Still Planning

Time for a travel plan update, though it's probably going to be short. But maybe I have some other things to write about.

The travel agent I'm going with has a subcontractor for local knowledge. In this case, a French woman. And I'm completely on board with that, but the sub didn't get back to her before Wednesday of this week. And that matters because the agent took off for vacation on Wednesday, so that means no further information until, at least, Thursday of next week. Probably later, because she's the owner of the agency, so she'll be coming back to utter chaos that she'll have to deal with first.

Or, you know, maybe she's a better business owner than I was, and everything will run smoothly in her absence.

Must be nice to go on vacation when you're a travel agent. You can get yourself the best deals.

And there's not much else I can do until I at least know when I'll be leaving, which I don't yet. I tried searching for fun things to do in France, but Google keeps pointing me at the Olympics, and I don't give a shit about that.

Which, it occurred to me, may be hypocritical of me, as I've harped in here again and again about how there's no such thing as useless knowledge or experiences. And I do believe that. My rationale (or at least rationalization to make me feel better about myself) is that I also tend to avoid anything that's too popular. Like Taylor Swift (who I acknowledge has talent) or "reality" shows.

But yeah, I do have an aversion to sports. You know how kids sit bored in math class wondering how they're ever going to use this? You ever notice no one says that about gym class, even though the vast majority of the kids aren't going to go on to be professional athletes? "But, Coach, when will I ever use sit-ups in my career as a veterinary technician?" "Well, um, strong abs will help you lift big dogs onto examining tables."

My argument is that you never know when the stuff they teach you in school will be useful or necessary. Was it "necessary" for me to spend high school in orchestra and drama class, let alone Latin? No, probably not. And I didn't know I was going into engineering until like senior year, so even math was questionable. But I learned all that shit anyway (and promptly forgot most of it).

The point being that just as strong abs are generally a good thing (something even I can acknowledge) even if you're not a pro athlete, so it is with a strong mind.

Now if only schools would emphasize that sort of thing instead of being all about the football team, we might actually get somewhere.

What does this have to do with travel? Well, nothing, except that maybe I can stretch (another good exercise) and claim that travel broadens one's perspective, so it's the same sort of thing. But, mostly, I'm rambling because I didn't accomplish anything else in terms of travel planning this week.

Next week, I'm going to look into relatively cheap laptops so I can bring one on the trip. Like a Chromebook or something, so I don't have to use my phone to make blog entries. I don't want to take my expensive gaming laptop with me (the one I'm typing on now), so this seems like a reasonable thing to do. Besides, it never hurts to have a backup here at home in case this thing blows a circuit one day.

And then I have to see about getting internet there. I'm well aware that they're ahead of the US in that regard, but that doesn't mean the systems are compatible, kind of like how I have a plug adapter to handle the different electrical outlet configurations.

5 Like • Comment • View 4 comments

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