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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10 Most Recent Blog Comments:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 22 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

March 11, 2024 at 8:26am

#1066054

Cheeses Crisis

Nooooooo! Someone needs to Do Something, Right Now!

Beware: A cheese crisis looms
Gird your curds! Say a prayer for Camembert! A collapse in microbe diversity puts these French cheeses at risk.

Just to get ahead of this: no, if this Vox article is correct, it's not because of climate change; yes, it's peoples' fault.

This looming cheese crisis, this Camembert calamity, stems from a much bigger problem: a collapse in microbial diversity.

On the plus side, Camembert Calamity would make a great name for a heavy metal Depeche Mode / Tears For Fears cover band.

Each hunk of Camembert or smear of brie is an ecosystem, an assortment of fungi and bacteria that turn milk fats and proteins into hundreds of different compounds. Those compounds produce the flavors, smells, and textures we love.

Hopefully, you already knew that the deliciousness of cheese is due to the action and presence of beneficial microorganisms. While we usually associate these things with spoiled food, they're all around us, and inside us, all the time; and as with multicellular plants or animals, the helpful ones outnumber the harmful ones.

Some are also responsible for beer and wine.

In recent decades, however, the genetic diversity of some of those microbes has caved.

The article will, of course, expand on this statement, but of course they gotta hook readers with the bad news.

To make cheese, producers typically take fresh milk and mix in bacteria and often fungi, including both yeasts and molds (fungi that tend to be fuzzy). Different microbe melanges produce different varieties of cheese.

It's obviously way more complicated than this, but the full process isn't really relevant to the story.

That's also a very simplistic description of mold... but an accurate enough one.

Historically, Camemberts and bries likely relied on mold strains from a species of fungi called Penicillium biforme, according to Jeanne Ropars, an evolutionary biologist who works at a lab affiliated with CNRS. Each strain was slightly different genetically, and so the resulting cheeses had slightly different colors, flavors, and smells.

I'm sure we all recognize at least part of that binomial.

Roughly a century ago, however, cheesemakers identified a particular strain of P. biforme that was fast-growing and albino; it produced a fluffy white mold that was, apparently, quite appetizing. This strain, known as Penicillium camemberti, was henceforth considered the gold standard for brie and Camembert (which differ from one another mainly in size). It quickly dominated the cheese industry, and the diverse group of other mold strains used to make Camembert and brie, and the colors they produced, vanished from disuse.

And, basically, that strain needs to be cloned, not bred, and cloning eventually results in DNA transcription errors.

The whole thing reminds me of the banana problem, which I've written about before, here: "Going Bananas"

Ironically, the problem there is one of the bad fungi (the kind that wear leather jackets, smoke cigarettes, and hang out in front of liquor stores). So it's analogous but not exactly the same thing. To its credit, the article mentions that.

This rapid caving of genetic diversity threatens other food industries, too, as the author Dan Saladino writes in his book Eating to Extinction: The World’s Rarest Foods and Why We Need to Save Them.

Hey, this was actually a book advertisement all along! That cheeses me off. At least be up front about it.

Ultimately, this doesn’t mean that we must bid farewell to brie, or that Camembert on toast is, let’s say, toast. There is a way to save these cheeses, though it requires some changes in our own taste and tolerance.

As for me, I look forward to greater cheese diversity. You want uniformity? Use that "pasteurized process cheese food" that America substitutes for culture (pun absolutely intended). As with beer and wine, I'd be happy to taste different cheeses produced with different friendly microorganisms.

As the Mandalorian might say, "This is the Whey."

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