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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September 25, 2021 at 12:01am

#1018010

Happy Mead-ium

I have had a complicated relationship with mead.

The Quest to Recreate a Lost and ‘Terrifying’ Medieval Mead
Bochet vanished for centuries, but meadmakers are bringing it back—at least in spirit.

Any sugar can be converted into alcohol. Feed it the right yeast, and magic happens. Well, actually it's science, but it might as well be magic.

For beer, first you have to convert the starch of a grain, usually barley, into sugar. That's easy enough, chemically. Saliva does that all the time, but of course these days other processes are involved. With fruits, though, such as apples for cider or grapes for wine, it's a more direct process.

Honey is basically just sugar, and it's used to make mead. But this mead is different than the kind you occasionally find in the stores.

It starts with a cauldron, an open flame, and a good measure of raw honey. Then—double, double, toil and trouble—stir constantly until the honey spits black steam at you. Add water and stand back as it erupts, volcano-like. Throw in some yeast and spices and, after it ages a bit, behold: bochet, a mysterious and lost style of mead.

Most mead is pretty straightforward. Not that I've made it myself, but I had a friend who did. I've had a few friends who created mead, actually, hence my complicated relationship with it: unless you do it right, it tastes more like ass than like honey. As for commercial mead, it also varies in quality. Oddly enough, in all of my travels to different breweries, I've only encountered one or two that have messed around with bee vomit to make mead. There's one in particular in a suburb of Minneapolis that I would totally revisit... but I digress.

This stuff, though? Bochet? I'd never heard of it.

“Caramelizing honey is kind of terrifying,” says Ontario microbiologist Bryan Heit, the brains behind popular homebrewing reference site Sui Generis Brewing. While Heit experiments mostly with beer, particularly traditional styles, he’s been intrigued by bochet for years.

Fortunately, I don't need to know how to make it when there are other people more than willing to do it for me.

“It truly is a lost style. It’s not a historical style that has survived into the modern era,” says Heit. “It’s literally something that disappeared.”

Why revisit something that's obviously died out? Well, why do anything?

Bochet is mentioned, briefly, in French texts as early as 1292, according to research by independent scholar Susan Verberg, published in 2020 in ExArc Journal. The first and only complete recipe for the drink—and the primary source for modern recreations—turns up in 1393, in what might be considered a manual of mansplaining.

The history of fermentation isn't well-documented, but fragments exist. I think I've linked the oldest known beer recipe in here before; it was from ancient Mesopotamia, and it wouldn't have resulted in anything much resembling what we know as "beer" in all its glorious styles. So in terms of history, 1292 C.E. is quite late in the game, but from a modern perspective, it's still old. (Mead itself is probably about as old as beer.)

I should note here that as accomplished as the French are at creating wine, their beer tends to suck on ice. While I have had decent French beer, it was all from the east, near places that actually know how to brew beer, such as Belgium and Germany.

So it's not surprising that they once made mead, although the fermentation of honey is more associated with the Norse. What would be surprising would be if it was any good.

A preserved recipe for an extinct beverage is a rarity, and its unique method tantalized historic reenactors and homebrewers alike, including myself. While I’ve made small-batch meads in the past, they have all been based on modern ratios, and none involved a cauldron. Commercial meadmakers have been particularly intrigued by the idea of caramelization: The process offers new possibilities, unlocking “all those roasty, toasty, nutty flavors,” says Jen Otis of KVLT Mead in Tacoma.

Caramelization isn't unique to honey; it's used on sugarcane to make molasses, which leads to rum -- but rum is a distilled spirit; if there are fermented beverages available from sugarcane, I'm unaware of them (this doesn't, of course, means that they don't exist). Likely they did, prior to the relatively modern process of distillation.

There are distilled spirits made from honey, too, but as far as I know, I've never tried them - just the mead, which as I've noted, can be of greatly varying quality.

The rest of the article, which is fairly long, goes into the author's re-creation attempts, and is quite fascinating, but there's no need to rehash it here. There's also the modern version of the recipe that she used. I'm not tempted to do it myself -- too much like work, and I don't really have the setup for it -- but if I ever see bochet available, you damn right I'm going to try it.

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