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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Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

June 16, 2019 at 1:16am

#960866

It's the Yeast We Can Do

https://www.atlasobscura.com/articles/cross-continental-origins-of-beer-yeast

Beer: Bringing the World Together.

The researchers found that Saccharomyces cerevisiae, the species of yeast that has been used to brew certain beers for thousands of years, is in fact a mixture of yeast strains found in both European grape wines and Asian rice wines. And so, they write, there would be no ales or lagers if not for the cultural exchange of fermentation technologies between Asia and Europe, possibly on the Silk Road.

There exists a sake brewery

here in my hometown now. I've been all over the country, and this is the first one I've ever seen. Naturally, I paid it a visit.

Sake is known as "rice wine" here in the English-speaking part of the world, but something about that never sat right with me. It's a fermented beverage, yes, like beer, wine, cider and mead. But the definition of beer is basically a fermented beverage made from grain, while wine is a fermented beverage made from fruit. Thus, cider is actually a wine... and sake could be considered a beer. Mead, being made from honey, which in turn is processed by sting bugs from flower pollen, is in a class by itself.

My trip to the sake brewery shed a bit of light on this terminology glitch. Wine tends to have a higher ABV (alcohol by volume, which measures the relative amount of ethanol in the elixir) than beer. The ABV of sake also tends to be on the high side, while the ABV of cider tends to be closer to that of beer. It appears that this is the source of the terminological confusion.

So, what controls ABV? Well, mostly, it's the yeast. As I understand the process, ethanol is produced by yeast when it processes sugar. In wine, that sugar is already in the fruit; for beer, a preliminary process converts the starch in the grain into sugar. Thus, anything with sugar or starch in it can conceivably become the source for delicious nectar - but some will be better than others. Barley is the go-to grain for beer; obviously, most wine is made with grapes. Some beers have other grains added. Some wines have other fruits added (I once had a pineapple wine on Maui, and it was delicious).

As the yeast performs its magic, eventually enough alcohol is produced that the yeast actually finds it toxic, and fermentation stops. It is also possible to halt fermentation at a desired alcohol level; this is done, for example, when local laws prohibit anything above a certain ABV. It is also possible to do things later to increase the concentration of the alcohol; taken to an extreme, you get distilled beverages such as whiskey or vodka.

An early stage of bourbon production, for example, is called "beer." This is not a particularly drinkable beer, and has not had hops introduced, but it's the fermentation product that will later be distilled into amber deliciousness.

So, basically, yeast eat sugar and pee alcohol, and when the pee builds up enough, the yeast starts to die off.

For the vast majority of the history of fermented beverages, people didn't know what yeast really was. It took microscopes to figure that out, and those weren't invented until the 17th century. By comparison, fermented beverages have been around for at least 10,000 years.

So... SCIENCE!

Thus, different strains of yeast can be isolated now, and some of them have a higher tolerance for ethanol than others.

The researchers sequenced the genomes of these strains—two ale, one lager, and one that contained both beer and baking strains—and compared them to a panel of every publicly available yeast genome from around the world, plus those from additional strains in their lab.

You're doing the holy work, scientists. Keep it up.

I should add, because the linked article doesn't really go into it, that there are basically two varieties of beer: ale and lager. The difference? The kind of yeast used. Each has its own preferred temperature. Ale lagers, as I understand it, are more tolerant of a higher alcohol content; therefore, ales tend to have a higher ABV than lagers. (Ale and lager each have many subcategories, of course, but those are the two parents of the family tree, as it were.)

So now I'm wondering what happens if you use a wine yeast for beer or vice versa. Can't be arsed to look it up right now, but I'm sure someone's looked into it. After all, it's a topic worthy of much study and contemplation, preferably after tipping back a few.

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