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

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

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Wordsmitty ✍️

about 12 hours ago
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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

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WakeUpAndLive️~Happiness

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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

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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"

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WakeUpAndLive️~Happiness

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

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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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November 11, 2023 at 8:49am

#1059300

Snack Time

Courtesy of Cracked, today we show how something built for laziness can get even lazier.

The 5 Laziest Snack-Food Innovations

I'm not knocking lazy, mind you. I'm all for convenience and instant gratification. They're two of the very few upsides of living in a declining civilization. These examples, however, are laziness on the part of the producers.

If you’re a snack company, what better way to rile up the populace than to reveal some sort of never-before-seen orientation of a preferred snack food.

You'd think they'd end up competing with themselves like that.

Sure, you’ve had Reese’s peanut cups for your whole life, but now that they’re shaped, in the vaguest possible way, like a pumpkin, they must be simply irresistible.

Nothing about Reese's cups is irresistible to me. I know this is blasphemy, as they're consistently at the #1 spot of favorite candies, but I don't particularly like them. Maybe if the chocolate were real, and the peanut butter wasn't dry and crumbly, I'd feel differently, but then they'd be expensive and no one else would buy them.

Oreos, now... I've been suckered into new Oreos fillings. Until they produced one that was ambrosia itself, and then discontinued them, at which point they (mostly) lost me as a consumer.

5. Oops! All Berries

I'll admit it: I ate Cap'n Crunch as a kid. It's legitimately the first breakfast cereal I have a memory of consuming. As I was very young, most of it probably ended up on the floor; I don't remember that part, though. And I ate the hell out of its variant with the "berries," but I don't recall craving just the "berries."

This actual product debuted way too late for me to have found out for myself.

I see through you with the ease of un-stained glass. You smelled blood in the water to the tune of sugar-addicted children for whom even the already heavily sugared default Crunch Particles no longer fed their fix. You get to debut a brand-new cereal, and the advertising push that goes along with it, without actually having to invent anything new. You’re just capitalizing on the human fascination of any new offering, and at the same time, hoping we don’t realize this is just a way for you to save on electricity and maintenance costs by giving the robot that combines the Captain Crunch and the Berries one day off a week.

And I don't believe for a second that the company cares about robots' rights.

4. Buncha Crunch

As far as I know, this candy is, despite the name, completely unrelated to Cap'n Crunch cereal... except for, possibly, the sugar content.

When Crunch bars debuted, they chose an unwieldy shape and presentation. A chocolate bar thin enough to be practically allergic to shipping, even before its base architecture was destroyed by a network of weak points in the form of puffed rice.

Worse, it's made by Nestle, which is the Platonic ideal of "evil corporation."

Merely spray chocolate wantonly over a conveyor belt of loose crunch and package them. Then throw them into a cardboard box that they couldn’t be bothered to toss an internal bag into.

Here, I think the author's giving Nestle too much credit. Seems to me they're just packaging their robots' mistakes. Also, still not real chocolate.

3. Corn-Flavored Doritos

Don’t act like some janitor at the Frito-Lay factory accidentally left corn in a toaster and now we’re being treated to a New World of Flavor. Do they truly think we’re no smarter than apes?

As the article notes, these are just tortilla chips, only unhealthier. Want corn chips? Get a hold of Fritos, one of that company's supposed flagship products. Their ingredients list is blissfully short, too: corn, corn oil, and salt. I don't believe in the mantra "if you can't pronounce it, don't eat it," because that encourages ignorance, but having four words of one syllable each can be appealing.

2 Crystal Pepsi

Not sure how this belongs on this list, but to me it's like: "How can we make a Pepsi that's even worse than Pepsi, without making a 'diet' version?"

1. Single-Stuf Oreo

Yes, I mentioned Oreos already, but this is a different issue.

Obviously, the original Oreo is the complete opposite of innovation.

Moreso because they ripped off the original creme-sandwich cookie, Hydrox.

Where my objection lies is in the fact that we cannot let them die the death they should have when their superior successor, the Double Stuf, took the throne.

No.

The Double Stuf is too much. The "creme" is not the best part of the Oreo.

I don’t care what you do with the old ones, market them Diet Oreos for all I care. Which I’m just cynical enough to think would absolutely work.

Stop giving Nabisco ideas. They have enough terrible ones

already.

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