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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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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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