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 8 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🎁

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

September 3, 2024 at 10:05am

#1076171

To Love Until I Laugh

"Blog Week Birthday Bastion 2024"

[13+] by WakeUpAndLive

Prompt 3. Sept 3.

You can achieve all the things you want to do, but it's much better to do it with loved ones around you; family and friends, people that you care about that can help you on the way and can celebrate you, and you can enjoy the journey. ~John Lasseter

Your most significant other(s) is/are....! Write about your loved ones (furry or not).

Right, as if I were capable of love.

Okay, okay, that's a joke, and I'm determined not to make today's entry as existentially terrifying as yesterday's. But as I showed then, you can't count on anyone. Just saying. And not just because I was given up for adoption as an infant, either.

Still, "love" carries a lot of baggage and assumptions. The Greeks might have gotten it closer to right by slicing it up into sub-loves: eros, agape, whatever. They say the Inuit have dozens of words for snow. While this is probably false, like all legends, it illuminates what might be a greater truth: the more important a concept is culturally, the more words they'll have to describe its various shades and gradations. We just have two words for love: love itself, and like.

We've got dozens of words for killing, though. And hundreds of words for penis.

All of which is to say that while there are people (and cats) that I particularly care about, I neither have nor want a "significant other." I've been married twice, and you know what Einstein supposedly said about the definition of insanity.

Einstein probably didn't say that, though, and it's not really the definition of insanity.

You know who (probably) did originate that quote, though? According to some of my fact-checking sources, in its current form ("Insanity is doing the same thing over and over again and expecting different results"), it was Rita Mae Brown, in the voice of one of her characters.

Rita lives near me. I met her a few times. Exceptionally nice lady. Cat person. I suspect she found it extraordinarily amusing that her quote became misattributed to the guy whose picture is in the dictionary under "genius."

Well, enough with the name-dropping. If it helps, I live near John Grisham, too, and never met him.

The point is, these days, I'm done with that sort of love. You know how some guys, they'll meet someone of their preferred gender, and their mind immediately fast-forwards ahead to the sex part? Mine fast-forwards to the breakup part, and I don't want to deal with that again.

But I have friends. And cats. That's enough for me.

Because if there's one thing I learned early and took with me into adulthood, it was how to be comfortable being alone. It always surprises me how much other people depend on each other, as if that rug won't be pulled out from under them at some point. Some say the pain of eventual abandonment is worth the pleasure of another's company, but I'm not so sure.

Don't dare think I'm bitter about it, though. I get that some people would be, but I'm not them. Mostly, as always, I'm amused, and laughter is one constant friend I can count on.

Let's do this again. We get to give out a free Merit Badge every day this week. Want one? Anyone who comments here before 11:30 pm WDC time today could get today's. (I'll need that extra half-hour to pick a winner and send the badge before midnight.)

To clarify:

When I say "comment," I mean comment. Not review. Though reviews are always welcome.

I also mean "here," not on the newsfeed post.

MB recipient will be chosen at random.

Maximum of one MB per commenter for the week.

If I don't get comments, I'll pick a previous commenter, and maybe not at random.

The MB will be the one I commissioned two years ago, "Complexity," which is a publicly available MB.

I appreciate all comments; this is just a little incentive.

2 Like • Comment • View 3 comments

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