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:

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.

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.

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

June 7, 2024 at 10:40am

#1072278

Pop Cuisine

Well, this is one way to get attention. And possibly draw more countries into WW3.

Croissants aren’t French and pizza sauce isn’t Italian – the national dishes that aren’t from where you think
A food historian has kicked up controversy after claiming that there is ‘no such thing’ as Italian cuisine, sparking debate over the origins and ownership of food. But perhaps we should reconsider our ideas about so-called ‘national dishes’, suggests Hannah Twiggs

In a new book called La Cucina Italiana Non Esiste (literally “Italian Cuisine Does Not Exist”), food historian Alberto Grandi claims, among other things, that Italians only discovered tomato sauce when they emigrated to the Americas, where tomatoes are native, in the 19th century.

It's trivially true that Europe in general had no idea about tomatoes until they started sailing back and forth across the Atlantic. That's hardly news. Reports of the fruit/vegetable's suspected toxicity may have been exaggerated, but clearly, at some point, they started eating them over there. The "tomato sauce" thing, though, I hadn't heard that before.

In an interview with the FT last year, he said that everything from parmesan and panettone to carbonara and tiramisu weren’t fundamentally Italian.

Okay, for various definitions of "fundamentally," I suppose. Are we going to take what they ate in ancient Rome, or Pompeii, and call that the only true Italian cuisine? Do we reject the idea of pasta as Italian because Marco Polo brought the idea for noodles back from Cathay? I mean, I don't know for sure if that happened, but my point remains.

Perhaps most controversially, he claimed that parmesan produced in Wisconsin was more authentic than Italy’s because it was closer to the original cheese produced in Parma-Reggio a millennium ago.

Ah, yes, the old "it's only authentic if it's done the old way" argument. Which is, generally, nonsense. Beer, for example, is a much more palatable product now than it would have been "a millennium ago," thanks to hops and science. Advances in the culinary arts, from technology, experimentation, imports, or just plain accident, contribute to the cultural landscape, not detract from it.

He was quick to point out, though, that he’s never questioned the quality of Italian food or products. “The point is that we confuse identity with the roots, which we are crossbreeding,” he told La Repubblica. “We wrongly talk about identity: cuisine changes continuously.” For example, the Italians and French are the biggest consumers of sushi in Europe.

As far as I can tell, everyone borrows or steals stuff they like from other cultures' cooking. Sometimes, it becomes so popular that it's identified with the new culture more than the old.

Case in point: Apple pie wasn't invented in America.

Michele Pascarella, owner of Chiswick restaurant Napoli on the Road – who has won countless awards for his pizza (eighth best in Europe) including being declared best pizzaiolo in the world last year – says it’s not about who does it first, but who does it best. “Italy is a country with an enormous food culture, passed down through generations, that doesn’t need to win any contest for who did it first in the world,” he tells me. “Our cuisine is envied all over the world and we even continue to make a difference today. Alberto Grandi is the flat-earther of gastronomy.”

I'm including this quote because "Alberto Grandi is the flat-earther of gastronomy" is one of the sickest burns ever ignited.

Nor do you hear protestations from the Japanese about tempura or the Indians about vindaloo. Both have Portuguese origins. Catholic missionaries brought the Western-style cooking method of deep frying to Japan in the 16th century, while vindaloo is derived from the Portuguese “vinha de alhos”, referring to the dish’s two main ingredients, wine and garlic.

I've written about this sort of thing before, when another source claimed there's no such thing as English food: "Food, Glorious Food"

With all that in mind, I have to wonder: does it really matter who invented tomato sauce on pizza? Or where your croissant is from? To acknowledge that one country might have had an impact on the food of another isn’t to be complicit in cultural appropriation. Point to any dish on a menu and you’ll have a hard time finding one that hasn’t got war, politics, economics, emigration or poverty to thank for its place there.

Not just the dishes, either. Here in the US, it's standard practice to keep salt and pepper shakers on the table. It's ubiquitous. Look up the origin of black pepper sometime. I've done that before, but I can't be arsed to find it again right now.

I'll just close today with this: one of the greatest benefits of living in a technoglobal society is that we get to pull from everywhere. All due respect to the well-meaning folks who urge "eat local," but there's no way I'm willingly giving up the benefits of being able to order tea from China, spices from India, hot sauce from Belize, beer from Belgium, wine from France, etc. (Yes, I know we have perfectly good beer and wine right here; it's the principle of the thing.) And those are just the tangibles; more importantly, we get ideas from all over the world. Some of those ideas suck, but some of them (like beer) make life worth living.

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