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:

Write_Mikey_Write!

about 8 hours 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

about 10 hours 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.

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.

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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