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 18 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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WakeUpAndLive️~Happiness

a day ago
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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

a day ago
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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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August 26, 2023 at 9:15am

#1054669

Ad Nauseam

This article is over 10 years old, but that's not a big deal because it's about marketing campaigns of the past that, in hindsight, might have been a bit questionable.

What Were We Thinking? The Top 10 Most Dangerous Ads

Turning everything into a "top 10" list may turn out to be a bit questionable, too.

For the full effect, you'll have to go to the link. There are pictures.

Often the criticism of vintage ads focuses on their inherent sexism, racism, or other displays of social prejudices, which we find laughable today, despite their continued presence.

"Laughable?" Okay, I'm hardly a perfect writer, and I don't always turn out to have used the most appropriate word, but I think "cringeworthy" might be closer to the truth here. Or possibly "horrifying."

But what about ads that steered consumers into dangerous territory, espousing outmoded scientific evidence or misleading half-truths to convince people that appallingly toxic products, or even deadly ones, were actually good for them?

And what ads of today, or of 11 years ago when the article was published, will be similarly disparaged in the future? I build my life around avoiding ads and commercials (historical ones like these are okay), so I rarely see any, but the practice of convincing people that something's good for them, when it isn't, is still going on.

But what about ads that steered consumers into dangerous territory, espousing outmoded scientific evidence or misleading half-truths to convince people that appallingly toxic products, or even deadly ones, were actually good for them?

I do think it's important to distinguish between ignorance and prevarication. Ignorance is when we just don't know enough, and might not even know we don't know enough. It can be forgiven, if later corrected. Outright lying, though, that's reprehensible.

So here’s a look back at 10 colossally painful advertisements, which make you wonder: What modern “health” products – vitamin water, granola bars, acai berry supplements – might look a little more evil in the future?

All of them, very likely. And anything "homeopathic," I'd hope.

1. Junk Food, Now Fortified with Vitamins and Minerals

Hm. No mention of Wonder bread or sugary breakfast cereals? Those were marketed under the veneer of "healthy" for way too long.

2. Let Them Eat Lead

This isn't even one of those where they could claim ignorance. (And no, "eat lead" here isn't a euphemism for "getting shot")

6. Plastics, Plastics, Everywhere

On the flip side, this may be excusable ignorance, heinous only in hindsight. At least... if it weren't for the whole "babies wrapped in plastic" thing, which I did, indeed, find laughable.

9. Shock Your Way to Physical Perfection

In 1922, “Violet Rays” were said to cure pretty much anything that ailed you. This Vi-Rex device plugged into a light socket so users could give themselves home shock-treatments, which would supposedly make you “vital, compelling, and magnetic.”

And this one outright cracked me up.

Various recalls and lawsuits erupted throughout the U.S., forcing the FDA to finally prohibit their manufacture. The last batch of Violet Ray products was seized in 1951.

For the mathphobes out there, that was nearly 30 years.

10. DDT For You and Me

And here's one I can't be arsed to figure out if it was ignorance or lying. But it's worth it to see the article just for the advertising graphic here.

The spread of DDT across mid-century America is mirrored today by the success of Monsanto (one of the companies that originally manufactured DDT) in placing its genetically modified products on store shelves before researchers have a full understanding of their larger ecological impacts.

Sigh. I'm glad they put this derpity-derp at the very end of the article, because had it been up front, I would have known I could safely ignore the whole thing. You know what they're going to look back on in the future and laugh? Well, lots of things, but the neuroses about GMO foods will surely be one of them. That's if we're able to feed enough people to have people to look back on us and laugh.

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