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.

3 Like

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.

2 Like

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.

1 Like

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.

3 Like

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.

2 Like

October 14, 2022 at 12:01am

#1039198

Tasting Red

Lies.

Everyone Likes Red and Pink Candies Best
Sweets manufacturers are finally catching on and selling packages without the lesser colors.

It only takes one data point to falsify the headline. Here it is: Me. I don't like red and pink candies the best.

Article is from 2015, but I doubt time has blurred the edges on the point. To mix a metaphor.

There’s an Internet meme floating around...that implores, “Don’t ever let someone treat you like a yellow Starburst. You are a pink Starburst.”

One of the reasons I live a solitary lifestye: cherry-pickers. If there's a mixed bag of treats, I take what I get (though I generally prefer the orange ones, or, in the case of chocolate, the dark ones). Kind of like how I enjoy choosing these articles at random. But everyone I know roots through the bag looking for choice morsels, which skews the balance and thus offends my sense of what's right and pure in the world. In a perfect world, my tastes and theirs would be different. But even though my taste isn't aligned with that of the majority, they still narf the ones I like best. Some of that, I suspect, is trolling. More likely, though, is that friends have similar tastes.

Except for the guy I know who mixes Skittles with M&Ms in a bowl. That guy's going to Hell for sure.

The message acknowledges and plays on a widely held belief: that pink and red candies are the best and all the other flavors are also-rans.

Okay, I can accept it's "widely-held." Don't generalize that into "everyone."

This isn’t to say there aren’t outliers, but more often than not, people prefer their fruity candy in shades of red.

So why the extremist, clickbait headline? Oh, right. Because it's clickbait.

It’s reminiscent of the frequency with which people claim seven as their lucky number.

I don't play craps, either.

Last year Popsicle came out with Red Classics, a line of its ice pops that nixes grape and orange from the usual lineup and only contains strawberry, cherry, and raspberry, an assortment of reds. Starburst has its FaveReds, a version of the fruit chews that includes strawberry (aka pink), cherry, and in place of orange and lemon, two rosy flavors: fruit punch and watermelon.

I don't eat sweets all that much, so I wasn't aware of any of these things. I suspect this article came to my attention because it's Halloween candy season. It might be fruitful (pun intended) to see if any of these things are still being marketed, eight years later, but I can't be arsed. In any event, while I don't dislike strawberry or cherry Starburst, I'm not a fan of fruit punch and I actively despise watermelon. I'd rather have yellow Starburst than watermelon. Hell, if there were a shit-flavored Starburst and a watermelon-flavored Starburst, and I had to pick one, I'd have to think about it.

Mogelonsky speculated that red was nonthreatening and lacked the acidic quality that can turn people off lemons and other citruses. But it’s not only that. The importance of the color red, sometimes over or in place of specific flavors, is notable. What is fruit punch, when you think about it, but a generic, noncommittal red flavor that doesn’t even bother to associate itself with a specific fruit?

What is fruit punch? Ass. It's ass. Watermelon is moldy ass.

According to Charles Spence, a University of Oxford psychologist who studies how people perceive flavor and consults for major food and beverage companies, color has a bigger influence on flavor than most people are aware. “There are probably a couple of hundred studies now since the first ones in the 1930s showing that if you change the color of a food or drink it will very often change the taste of the person rating it,” Spence said. “You sort of think intuitively, well … the color isn’t part of the taste. And yet this growing body of research over the decades does show it can influence the taste in quite dramatic ways that can’t necessarily be overwritten.”

That's because taste isn't everything. It's the whole experience that matters. Nowhere was this more evident than during the Great Coke Crisis of 1985, a truly dark time in our history. Blind taste tests apparently indicated that people preferred the disgusting taste of New Coke to the tried-and-true formula. What the marketing gurus didn't seem to understand was that we were used to the taste of actual Coke, and had grown to like it. They learned. Eventually.

Speaking of Coke and red, a couple years ago Coke changed the packaging of their "Zero" product (which is what I drink now when I'm not drinking the harder stuff) from black cans to red. A slightly different shade than Coke/Santa red, but still red. I wonder if that increased sales, given the apparent preference for red, but again, I can't be arsed to research it. I didn't care, because it still tasted the same.

This "whole experience" thing is why you don't do blind taste tests of beer or scotch. I'll be the first to admit that I drink expensive scotch because it's expensive and I can. Are there cheaper whiskeys that are just as good? Probably. I don't care.

Regarding red, Spence said that studies have shown that “it seems like red is a particularly effective cue for sweetness, maybe because there’s a cue in nature, which is fruits going from green and sour and unripe through redder and sweeter and riper.”

A moment's thought should be enough to falsify that hypothesis. First, lots of sweet fruits aren't red. Oranges, e.g., or blueberries (it's right there in the names). Second, lots of red things aren't sweet. Like tomatoes.

Marcia Pelchat, a psychologist at the Monell Chemical Senses Center who studies food preferences, disagreed with Spence’s theory. “I don’t think you can make an evolutionary argument that goes back to our primate ancestors. I think it’s shared cultural experience,” Pelchat said.

Another strike against evo-psych.

Confection and dessert companies are certainly aware of the power of red. “For our brand, red is a magical color,” said Nick Soukas, who oversees ice cream products at Unilever, the owner of Popsicle.

Maybe it's because meat is red, and meat is delicious.

As for me, I associate red with holidays I hate.

Spence agreed that anything novel can grab attention and sales. And in an era where you can custom-order any color of M&M’s online, in colors that used to be strictly unavailable, maybe other candy companies feel they need to flood the market.

Like I said, I don't eat much candy these days. But when I do, color is the least of my concerns.

As long as they're not watermelon flavored.

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