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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QPdoll

about 7 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 10 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

about 21 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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December 29, 2023 at 7:44am

#1061544

I Don't Know, But Alaska

Ever read something just to remind yourself how good you've got it? For me, that's pretty much everything from Outside, especially this article:

The Joys of Cabin Living in Alaska
Want to know what domestic bliss looks like? A rundown cabin with no electricity on the edge of rain-soaked Alaskan wilderness.

Huh... I didn't know that "domestic bliss" was a synonym for "literal Hell," but in retrospect, it makes sense.

Article is from 2013... not enough time for climate change to significantly affect the subject matter or my feelings about it. Yet.

My two brothers and I, along with a buddy of ours... own a shack at a place called Saltery Cove on Southeast Alaska’s Prince of Wales Island.

Morbidly curious, I looked it up. It's almost as south as you can get and still be in Alaska. The northern boundary of British Columbia is quite a ways north.

The place is still way too north.

The shack is about 36 feet long and 12 feet wide, with the warped shape and discoloration of a cardboard shoe box that’s been soaked in the rain.

Luxury!

The appeal of our shack isn’t so much the structure itself, but rather the bare-bones nature of its locality. Surrounded largely by the Tongass National Forest, it’s a place where black bears gnaw mussels from the rocks in what might be described as our yard and killer whales pass by so close that you can hear them even with the door closed.

Okay, fine, that would be kinda cool to experience. For, like, a day or two.

But in truth that’s only half the answer. The other half is more difficult to explain and also a bit masochistic: Saltery Cove is a place where everything—the weather, the ocean, the mountains, the people, the trees, the animals, even the buildings—seems capable of kicking your ass in a very physical way. And in today’s increasingly tame and virtual world, where our primary sensations tend to be delivered by our Wi-Fi connections, a good old-fashioned ass kicking is something worth paying for.

Bullshit.

I mean, okay, you feel that way, fine. We gotta have differences of opinion, or else one spot would get too crowded. But to me, this is like saying, "I built myself a beautiful mansion, but I prefer the feeling of living in a drainage ditch."

I prefer the simple life. By which I mean, if I'm cold, I press a couple of buttons, and then I'm not cold. If I'm too hot, I press the same buttons in a different order. Hungry? Buttons. Thirsty? Okay, no button for that, but there's a lever. Dirty? Knobs for that. Entertainment? Back to the buttons. Couldn't possibly be simpler, at least not until we further refine AI. No, I don't yearn for the complicated lives of my ancestors; they worked their asses off so I could sit on mine all day, and I'm ever so grateful and appreciative of them for that.

Another way in which the cabin kicks my ass is through my wife, Katie. She often regards my purchase of the shack with that eye-rolling sense of dismissal that people will use when confronted with the subject of their spouse’s past girlfriends or boyfriends.

And yet she puts up with you. I've never really understood people, and articles like this only add to my confusion.

I went there with my brother Danny to fish salmon and halibut with one of Saltery Cove’s eight full-time residents, Ron Leighton, a man of mixed Native Alaskan and Irish descent who’ll tear your head off for tangling an anchor line and then send your kid a birthday present even though the nearest mailbox is an hour’s boat ride from his house. Ron’s résumé includes a tour of duty as a door gunner in Vietnam, a career as a detective with the police force in Ketchikan, Alaska, and a parallel career as a halibut long-liner.

Gotta admit, though, that's the kind of person whose life seems tailor-made for writers.

The article goes on to describe why it's such a wonderful idea to buy a cabin on the coast of nowhere, sight-unseen: in summary, the place was utterly trashed and they spent months cleaning up.

Oh, well. It works for them. More space in semi-civilized areas for me, then.

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