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🎶

about 21 hours 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.

1 Like

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

July 21, 2024 at 8:39am

#1074214

Revisited: "Smiley"

My weekly trip into the archives only took me back two years, this time, to July of 2022, almost to the day. It's a fairly long set of musings on a fairly long article from aeon: "Smiley"

And it is, as one might guess, all about smiles.

Now, it's only been two years, as compared to the many years of history discussed by the article. One of the reasons I do these things is to see how things have changed, and, well, not much has changed in two years. There's some discussion of face masks, which were more prevalent in public in 2022, but that's about it.

I will, therefore, just point out a couple of things I either missed or left out in a failed attempt at brevity back then.

First, as academics are wont to do, the article includes a foray into Latin:

Ancient Romans showed another variant. If we take their vocabulary at face value, they did not distinguish between a smile and a laugh, contenting themselves with a single Latin verb – ridere – for both. Only towards the end of the Roman Empire did a diminutive – subridere – enter the language. This came with the derived noun sub-risus (later, surrisus) a ‘sub-laugh’– a little or low laugh – associated with mockery. It retained this lesser status and this diminutive form, distinguishing it from the laugh as it entered the Romance languages in the High Middle Ages. Around 1300, for example, French contained words for laughing (rire) and laughter (le rire or le ris) and smile (sourire, from sous-rire).

I'll point out now that the Latin ridere is the known origin of our words "ridicule" and "ridiculous," which today don't conjure up much in terms of smiley laughter, but rather lip-raising scorn, such as I occasionally heap upon stupid articles in here. Not the one under discussion today, but others.

Second, it's been bandied about that forcing a smile also forces the underlying emotions of happiness, contentment, or relaxation, and supposedly makes one appear more friendly and approachable.

Neither of these things is always good, in my opinion. Sometimes it's best to work with the emotions you're actually feeling (or so I've heard, at least for actual humans). And one does not always want to look friendly and approachable, because someone might approach you and try to get friendly.

It could even be a symptom of the foul practice of toxic positivity.

And to me, forcing a smile is the exact opposite of relaxation. I'm not suggesting frowning as an alternative, but just... letting it be. Unless you're in the kind of situation where people shouldn't know what your actual emotions are, lest you get intrusive questions like "Why the long face?" or "Okay, what are you grinning about?" Or if, you know, you're at a poker table.

I still pity the retail workers who have to go all day with a fake smile plastered upon their face. Even worse are the ones who are genuinely happy to have to present their customer-service face to the world. I mean... why? Smiley faces are for people who aren't paying attention.

So anyway, the original entry and article are there at the link, and I still think the article's interesting. But I don't think I smiled even once.

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