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 21 hours 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

a day 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🎁

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?

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

August 11, 2024 at 9:14am

#1075049

Revisited: "It's a Trappist!"

Well, this entry from July of 2020 is embarrassing: "It's a Trappist!"

For starters, it wasn't the first time I'd used that title. In my defense, it's a truly awesome pun combined with a Star Wars reference joke, so I tend to use it a lot.

The entry itself was a response to a 30DBC prompt that went like: "If you won a free trip to any foreign country, all expenses paid in your own private jet and had the time to go (and there were no travel restrictions due to a global pandemic

), what is the first foreign country you would visit? Who would you bring with you? What would you spend your time doing?"

And I think we all know the answers to those questions as of 2024, but four years ago, things were a bit different.

Last year was my mostly-stay-at-home year. Not only did I want to conserve money for upcoming trips, but I wanted to lose weight so I wouldn't look like a typical American.

I was mostly successful at both.

I got even better at conserving money for upcoming trips because I could barely go anywhere for four years. The weight thing, I don't talk about.

The plan was to go to Scotland in May with a friend for the Islay festival.

That might still be my next venture. Obviously, it hasn't happened yet.

This shit will not be resolved by next year. So I doubt I'm going to Belgium or France. Or Scotland. Or the Netherlands. Or Belize. Or even travel in the US.

Oh, but I did end up doing a road trip in the US the following year. That did not end well (though it could have been a lot worse).

I'm prone to depression and this shit is not helping. Not the pandemic, not the travel restrictions caused by the pandemic, and certainly not prompts like this one. The only thing I have to look forward to these days is travel and death, and it looks like the latter is going to happen before the former.

See, this is why I'm a pessimist. Either I'm wrong, which makes me happy, or I'm right, which makes me happy. If I were an optimist, I could only be disappointed by the results.

Even if I somehow manage to stay alive and reasonably healthy (unlikely), my passport will probably expire before I get a chance to travel again. And I haven't even used it since I last renewed.

The major impetus behind planning a Europe trip this year is to use that damn passport before it expires.

But right now? In the utterly impossible scenario above? Belgium. With my friend, as I mentioned, and whoever else wants to go. Primarily, I'm interested in immersing myself, at least figuratively, in Belgian beer. Trappist ales, saisons, whatever. I can get some of them here -- I just had a nice couple of bottles of Delirium Nocturnum yesterday, and there's still one in my fridge along with a big bottle of Kwak -- but as with California and wine, they keep the really good stuff for themselves.

Coincidentally (or not), I have some Delirium Nocturnum in the fridge right now. It's very difficult to find Kwak here.

So the rest of that entry was just responding to the prompt. I'll be traveling alone, and also visiting France, so, not exactly what I had planned (which I don't mind), but dammit, at least it's (probably) going to happen and I can stop annoying everyone with "what I wanna do" posts.

3 Like • Comment • View 2 comments

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