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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March 20, 2021 at 12:01am

#1006689

Fish and Ships

Just some fun historical trivia today.

How Fish and Chips Migrated to Great Britain
The fried fish was introduced by Jews fleeing religious persecution.

I could list a few dishes that became generally popular due to Jews fleeing religious persecution. The Irish-American penchant for corned beef and cabbage on St. Partick's Day, for starters.

The powerful pairing of fish and chips has long been considered a British staple. Dubbed “the undisputed national dish of Great Britain” by the National Federation of Fish Friers, it’s been enjoyed on the island for over a century, with an estimated 35,000 chip shops in business by 1935.

I made sure to eat plenty of fish&chips when I was in England, though I was unaware of its historical origins. The best I had was in this little town on the east coast called Whitby, where they go out and catch the fish in the morning so it's all ready for lunch.

From the 8th to the 12th century, Jews, Muslims, and Christians lived in relative peace in Portugal, known as Al-Andalus under Moorish rule.

I'm guessing that was the first and last time.

As religious violence worsened, many fled Portugal and resettled in England, bringing with them culinary treasures founded in Sephardic cuisine—including fish.

There were two main branches of European Judaism: the Ashkenazim in the east, and the Sephardim in the west. Culturally, they were quite different. It was mostly Ashkenazim who immigrated to the US, bringing with them such things as corned beef, bagels, and latkes.

The dish of white fish, typically cod or haddock, fried in a thin coat of flour, was a favorite particularly among Sephardic Jews, who fried it on Fridays to prepare for the Sabbath, as the Mosaic laws prohibited cooking.

Hence why it's called "FRY day."

That's a joke. Friday was named after a Norse god, like most of our days of the week.

But the Friday-night tradition was likely chipless until the late-19th century. The general popularity of the potato bloomed late in Europe, and it wasn’t until the late 1800s that the tuber was accepted, due especially to the promotional efforts of a French scientist.

And yet what we call French fries and the British called chips were, if my sources haven't failed me, a Belgian invention.

There are also competing theories about who created the pairing of, as Churchill called them, “good companions.” Most trace it back to the early 1860s, when Joseph Malins, a Jewish immigrant, opened up a fish and chips shop in London. Others point to John Lee, a man living outside of Manchester, who ran a “chipped potato” restaurant that sold the beloved pairing.

Because London and Manchester need another excuse for a rivalry.

British natives and immigrants alike began slathering their cod in batter and frying up husky chips. Industrialization in the 19th and early 20th centuries launched the fish dish to even greater heights, as it became a favorite for factory and mill workers in London and beyond.

One of the great things about food, besides being necessary for continued existence, is how different cultures adapt and expand on other cultures' cuisines. Some call this cultural appropriation. While that's not impossible, for the most part, I see it mostly as a beneficial thing. I've mentioned in here before that certain Italian cuisines, based off of pasta and tomatoes, couldn't exist without noodles from Asia and tomatoes from the Americas. The potato itself, of course, was a New World plant as well.

We live in a time when we can sample foods (and booze) from all over the world, and as far as I'm concerned, that's a wonderful thing.

What really matters with food is not where it came from, but how delicious it is. And it's hard to get more delicious than fish and chips.

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