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 9 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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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 2, 2024 at 1:35am

#1076064

To Live Until I Die

"Blog Week Birthday Bastion 2024"

[13+] by WakeUpAndLive

Prompt 2. Sept 2.
Tell us about an earthshaking Life-or-Death situation in your life. What happened, how did it change you, if at all?

I've been close to death three times that I know of, each time from some medical issue.

I say "that I know of," because who knows what might have happened had I crossed a street a few seconds later, or decided to go out instead of staying home?

Thing is, none of those instances were particularly "earthshaking." For a human over the age of, maybe, 3 or 4, being alive means living with knowledge of your own mortality, even if only in abstract terms. As better writers than me have noted, death is one of the very few certainties, along with taxes and construction on the Capital Beltway. Doesn't make it easier to contemplate, maybe, but you know it's coming.

And with age comes even deeper knowledge of the inevitability of drawing one's final breath. You've lost many pets, if you're lucky enough to have had pets. You've lost family. Maybe you've lost friends to the Reaper's scythe. Or, like me, you watch your parents slowly decline into husks of their former selves, to the point where death becomes a release and a relief, not something to be feared. It creeps closer and closer, as sure as getting honked at in Manhattan. The only unknown is: how are you going to handle the knowledge?

Back when Facebook was new and shiny, I had an account there (still do, because I can't get into the damn thing to close my account, but that's not really relevant) and I, like many people, became curious about what people I knew long ago were up to. I found a mutual friend of a high school girlfriend, whom I asked about—not trying to start anything back up, but like I said, just curiosity. "Oh, she died a few years ago," the mutual friend told me.

Apparently, she'd just gotten back from her honeymoon and was in great spirits, looking forward to her new life, when she collapsed on a sidewalk from an undiagnosed congenital heart condition. She was in her early 30s. Her twin brother, who, somewhat predictably, never much liked me, got checked out; turns out he had the same condition, but knowing about it, the docs were able to treat it. Last I heard, he was still kicking. I even ran across his name in a news story once, a few years back.

I may be getting some of the details wrong, because this was long ago, and her death even longer ago, but the result remains the same: that's most likely when it really hit me that anyone can die of anything at any time. Sure, I'd known that intellectually, but that was probably the moment when I felt it in my gut. What changed my perception, at least to some degree, was facing not the prospect of my own death, but getting hit by the news of the death of someone my own age I'd once cared about.

Some people deal by trying everything they can to extend their life. I can understand that, though that's not me. Other people deal by deliberately trying to shorten it. That's not me, either. Still others cling to the belief that there's a life after death, that their consciousness will, through supernatural means, survive their body's demise. While I'm not here to disparage anyone's closely held beliefs, I couldn't take comfort in that; one of the many things I'd find scarier than death would be eternal life.

No, between watching my parents decline and die, and hearing about my former girlfriend, the only thing that makes any kind sense to me is to live while I'm still alive. Sometimes, that means doing things that have been shown to reduce one's life span, but that's not why I do them. I do anything I do for one overwhelming reason:

I fucking feel like it.

And that, folks, is why I plan to be in France three weeks from now: because I feel like it.

I could die tonight. Or on the trip. Or 20 years from now, or anytime in between. But I lived first.

Life's too short to drink the cheap stuff.

You wanna live until
You die alone and will
And I can fly alone at will
I'm not so far below
I'll live beneath your sky
With tainted eyes
I've made my mind
To live until I die

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.

5 Like • Comment • View 7 comments

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