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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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').

1 Like

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

JayNaNoOhNo

Nov 5, 2024 at 1:16pm
In response to "Anomalies"

*Toronto, Canada submits a sternly-worded petition and disputes the center of the universe claim.*

2 Like

March 9, 2023 at 9:23am

#1046131

Saison de Cueillette

It's time, once again, to talk about one of my favorite subjects.

The Brewer Updating an Ancient French Beer for Modern Drinkers
Thomas Deck’s cervoise-inspired brew is herbal and delicious.

Consider this:

Belgian beer is, generally, delicious.

British beer is, generally, delicious, though very different to Belgian beer.

French beer is, generally, ass.

I'm certainly not ragging on France, here. When it comes to wine, they're exceeded only by California (oh yeah I went there). But apart from the products of a few breweries near the borders of Belgium and Germany (which also makes fine brew), every French beer I've had has been disgusting.

But. My experience is limited, and things change...

But ancient Celtic inhabitants of modern France likely did indeed drink cervoise, an unhopped beer sometimes referred to as a gruit in English and whose consumption dates back to antiquity through the Middle Ages.

The prevalence of hops as an additive to beer, and then, later, an absolutely essential part of the holy nectar, is relatively recent, compared to the long history of beer itself. So it's no surprise that some of these early, Roman-era malt beverages did not use hops. (Hops, by the way, are a preservative, extending the life of a beer—essential in pre-refrigeration times. They also add a bitterness that nicely offsets the sweetness of the malt and alcohol.)

Traditional cervoise recipes, according to Malcolm F. Purinton, a professor who teaches the history of beer and alcohol at Northeastern University...

Nice job if you can get it.

...were likely sweetly herbaceous, infused chiefly with a trio of yarrow, wild rosemary, and sweet gale or bog myrtle—though they could also include other herbs and spices, depending on what was available.

Yeah, should have stuck to wine.

This year, Thomas Deck, a brewer based in the Paris area, set out to recreate a cervoise for the contemporary Parisian. Though he made a few modern updates for stability and flavor, “it’s a little bit inspired by, like, druids cooking something in a cauldron, and throwing herbs into it,” he says.

And we know druids did this because...? Well, never mind. Whatever the origin, it's beer, and I want to try it.

Deck could well have been destined for a brewing career from early childhood. He grew up in eastern French Alsace, which brews around 60 percent of all French beer, according to Visit Alsace. But it took a chance encounter with American home brewer Mike Donohue while studying abroad in Washington, D.C., for Deck’s eyes to be opened to the world of craft beer, then nearly absent from the French scene. Ten years later, in 2014, he and Donohue founded their own craft brewery—Deck & Donohue—in the Parisian suburbs. It was one of the first in a land that, to hear Deck tell it, had barely heard of IPA.

Yes. Alsace. Right up next to Germany. Like I said.

Incidentally, the French having barely heard of IPA was a point in their favor. Of all the styles of beer available, that's among my least favorite.

In the years since, Paris’s beer scene has witnessed a veritable explosion of craft brewers, many of whom lean hard into local sourcing, branding themselves with that oh-so-French word more frequently attributed to wine: terroir.

Eh? Really? Shit. Now I have to visit Paris. Oh no.

“Herbal is tricky because if you overdo it, it’s really disgusting,” he says. “You want it to be noticeable, but you don’t want it to be like … feeling like it’s an air freshener product.”

This guy gets it.

For his cervois experiment, he made the choice to deviate from historical accuracy and use a touch of Alsatian Nugget hops for stability and their bittering properties, boiling them to ensure that their aroma didn’t interfere with the flavor of the herbs. The result is more of a modern saison than an authentic cervoise, but Deck considers this an essential compromise.

There's a reason some things fall out of favor; it's generally because something new and better comes along. In this case, it was hops. So I see nothing wrong with adding a little to make it palatable and stable, if less "authentic." Especially since they're not sure what "authentic" really means.

This year, he has brewed just 2,000 liters of his cervoise-inspired beer. It’s fresh, herbal, and delicious, a simple, low-ABV, easy-drinking beer. If you didn’t know its backstory, you’d have no idea so much philosophy and research had gone into it. This rustic simplicity is reflected in its very name: Saison de Cueillette, or forager’s saison.

Yep... if this war ever ends, I'm going to France in addition to Belgium.

Tant de bières, si peu de temps. (So many beers, so little time.) I should make that my motto.

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