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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Writer_Mike

about 11 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

a day 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.

2 Like

Annette in France

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

Bienvenue en France.

1 Like

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

1 Like

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

1 Like

Elisa: Middle Aged Stik

Sep 21, 2024 at 7:34am
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

2 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:46am
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/

3 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

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

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
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.

February 28, 2023 at 11:20am

#1045698

In Your Face

After yesterday's foray into the deep, murky, turbulent waters at the confluence of science and philosophy, how about a little comic relief, from Cracked?

Ways Foodmakers Get Us To Buy Overpriced, Trendy, 'Healthier' Foods

Well, let me guess: with apologies to Mark Twain, there are lies, damn lies, statistics, and marketing.

Trendy food-makers, as you've undoubtedly noticed popping up in unprecedented numbers and diversity recently, have two goals: 1) relieve you of your hard-earned funds, and 2) make decent-tasting, more healthful foods. And, sadly, often in that order.

Hence why they nearly always fail at #2. I find that people don't actually want good-tasting healthy food. If it tastes good, it must be bad for you, so you have to do penance. Thus: kale.

Like non-GMO? Literally, most modern fruits and vegetables were the product of GMOs, and have been since before any of those words existed.

Thanks, Cracked. I've been saying this for years.

And gluten-free? Gluten is a protein that forms when flour is mixed with water, then kneaded.

I think gluten itself has a marketing problem. The problem started with chemists, who called it that to begin with (and make no mistake, all foods are made up exclusively of chemicals; you're not going to find chemical-free food anywhere in this universe). The word "gluten" and its derivatives are all over chemistry; for example, monosodium glutamate. The marketing problem is this: the word resembles "gluteus maximus," which is Latin for ass. So your mind goes "it must taste like ass." Which it doesn't.

Now, I know that there's a relatively small number of people who legitimately have a gluten allergy, and it's helpful for them to have those labels. But the marketing comes in when food companies make no effort in saying "this product contains gluten, which is actually quite tasty and not a problem for over 99% of you," instead labeling something "gluten-free" as if that's some sort of health benefit.

But trendy food-makers also like to make you think their snack foods, especially chips and crackers, are less fattening than their run-of-the-mill, blue-collar competitors.

Sure, and once you label something as "healthier and less fattening," people will eat more of it. Which of course is the whole point of marketing, but it's no better for the consumer to eat two bags of 500-calorie crisps than to eat one bag of 1000-calorie crisps. And yet:

But once you check the nutritional info, you'll see that the calories are nearly identical to regular potato chips. Additionally, the vitamin and mineral profile isn’t much better.

It's almost as if one would be better off eating an actual vegetable rather than a starch (which is a carbohydrate and gets converted into sugar).

Not to mention the glut of gussied-up tortilla chips, with flashy labels and prominent boasts of "red quinoa" and "multigrain" and other buzzwords to conceal that you're eating a corn chip with a sprinkling of birdseed tossed into the industrial mixer.

I'm just leaving this here because it made me actually LOL.

I have a thing for Frito's. Ever see the ingredients list on Frito's? "Corn, corn oil, salt." That's it. No preservatives (apart from the salt), no monosodium glutamate (not that there's anything wrong with that), and not a single polysyllabic chemical word.

I'm not saying they're good for me. I'm just saying they're pretty simple.

Other trendy treats will assure you that they're "naturally sweetened" and that their cookies, bars, or balls eschew evil sugar for the inherent, wholesome sweetness provided by dates or coconut. Sounds great, except there's not much difference, especially since dates and coconuts are among the most calorific things you can stuff in your mouth.

While this article focuses on calories, I'm aware that there are other measures of the "healthiness" of a food. What they are, though, is still being debated, and I don't trust nutritional science as far as I can throw it (mainly because much of it is funded by these same food companies, so there's an incentive to get the results they want).

Plus, though many will hate reading this: sugar isn't evil. So the best way to chisel at those love handles is to track our total caloric intake and not worry so much about the source. Or to make a soul-pact with Lucifer; your choice.

Jury's still out on that, too. My understanding is that white sugar is simple carbs (bad) while brown sugar is more complex carbs (not as bad). But who knows what they'll figure out next? Meanwhile, I'm not going to worry too much about it, but, as always, I'll be aware of marketing gimmicks.

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