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 31, 2024 at 8:37am

#1075949

Revisited: "Meet George Jetson"

In August and everything after
You get a little less than you expected somehow
—Counting Crows

I usually do these archaeological expeditions on Sundays, but, as I noted yesterday, all of next week will be (or at least should be) taken up by entries for "Blog Week Birthday Bastion 2024"

[13+].

The past, imperfect, goes all the way to April of 2007 today, for a short entry about consumer automation: "Meet George Jetson"

My wife wants a robot.

Yes, this was long enough ago that I was actually married.

No, not for that. I haven't been replaced by a machine yet. Or, well, maybe I have; her friends have been holding a lot of schtupperware parties lately.

"Schtup" is a Yiddish word for sexual intercourse. Her friends had these get-togethers where the host would demonstrate various "toys" (mostly made of plastic) and get some sort of commission on sales. It would have been impossible for me not to make a pun on Tupperware parties.

By "demonstrate," I don't mean what you're thinking. Get your mind out of the gutter, please.

But specifically, she wants a robot she saw that travels around the house, cleaning it.

A few years back, my housemate got a Roomba. It was a massive pain to organize around, since neither of us is naturally organized. There's barely enough open floor space for it to do its thing, and I can't be arsed to do all the work it would take to possess less stuff. It collects dust now, on top rather than inside.

Now, overlooking for the moment the years of therapy our cats will have to endure if we actually get this little slice of science fiction...

Also, the cats were not amused. Yes, I've seen those videos of cats sitting on the damn things while they scoot around, but my cats are different.

...glossing over the price (which, really, is probably comparable to an Oreck vacuum cleaner)...

The house is, and was, uncarpeted except for one single room. Not worth spending that kind of dough. Then I'd be the sucker instead of the vacuum cleaner.

...and even forgetting for the moment how this little gadget confirms my theory that if necessity is the mother of invention, laziness is the milkman...

Huh. I've been saying that for at least 17 years. I had no idea.

...aside from all that, there's something fundamentally wrong here:

There exists a household cleaning robot, and I still don't have my flying car.

Every time I complain that I don't have a flying car, two responses come soaring in, as if jerked by knees:

1) Flying cars do exist (generally accompanied by a link to a video);

2) People can't drive in 2D; imagine the chaos if they tried it in 3D.

For 1, I say "Okay, they exist. Fine. They're not production models and, more importantly, my complaint is not that they don't exist, but that I don't own one."

For 2, my response is that I don't give a good goddamn if anyone else has one; I'd be perfectly happy being the only owner of one.

I was promised a flying car. It was right there at the 1939 World's Fair: Flying Cars are the Future.

The far future, apparently.

Point is, how come she gets a household robot, and I'm still stuck driving a vehicle that never leaves the ground (except maybe when I'm being chased by Roscoe P. Coltrane)?

Spoiler: she didn't get a household robot. I'm not saying that the lack was a proximate cause of our divorce, but it probably didn't help.

There follows a link to a Popular Mechanics article (I presume from the URL), but the link doesn't work any more and I have no idea what point I was trying to make with it. Nor can I be arsed to search it out.

And just to forestall another assumption: it's not like I expected her to do all the cleaning. We both did that. Not that I would have done as much if I'd been single, as evidenced by the fact that these days, I hire a maid service, which is almost as much an organizational pain in the ass as the Roomba was.

2 Like • Comment • View 1 comment

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