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.

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.

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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.

2 Like

February 17, 2021 at 12:02am

#1004657

Attack of the Killer Tomatoes

You say toe-MAY-toe, I say toe-MAH-toe...

Why the Tomato Was Feared in Europe for More Than 200 Years

Clearly, it was because no one knew how to pronounce it. Hey, let's mess with everyone's heads and start pronouncing it toe-MATT-oh. Who's with me? While we're at it, we can restart the old argument over whether it's a fruit or a vegetable (truth: it's either, depending on whether you ask a botanist or a chef).

In the late 1700s, a large percentage of Europeans feared the tomato.

To be sure, there was a lot to fear in late 1700s Europe, and not just if you were a French noble.

A nickname for the fruit was the “poison apple” because it was thought that aristocrats got sick and died after eating them...

Which is how the tiny violin got invented.

...but the truth of the matter was that wealthy Europeans used pewter plates, which were high in lead content.

And they probably ate mercury off of them, too. No, really: apparently mercury was considered a cure for syphilis. This is one reason why I keep saying the past sucked.

Because tomatoes are so high in acidity, when placed on this particular tableware, the fruit would leach lead from the plate, resulting in many deaths from lead poisoning. No one made this connection between plate and poison at the time; the tomato was picked as the culprit.

This is why science is important. You need a double-blind test. Give 1/4 of the aristocrats pewter plates with tomatoes on them. Give another 1/4 of them pewter plates with placebos on them. The third 1/4 gets placebo plates with tomatoes, and the remainder get placebo plates and placebo tomatoes. Record how many of each group dies. Then guillotine the survivors just in case.

Around 1880, with the invention of the pizza in Naples, the tomato grew widespread in popularity in Europe.

Cultural appropriation! The tomato isn't native to Italy, so it's inauthentic cuisine! Oh, well, we got 'em back when New York perfected the pizza.

Before the fruit made its way to the table in North America, it was classified as a deadly nightshade, a poisonous family of Solanaceae plants that contain toxins called tropane alkaloids.

This, in spite of European invaders literally seeing Americans eating the things.

Like similar fruits and vegetables in the solanaceae family—the eggplant for example, the tomato garnered a shady reputation for being both poisonous and a source of temptation.

To be fair, eggplant isn't actually food. Oh, sure, I understand that some people eat it, but I refuse to classify it as edible. Consequently, it is not a source of temptation for me. Neither is the tomato, which I can take or leave, but at least I acknowledge it as food.

Gerard’s opinion of the tomato, though based on a fallacy, prevailed in Britain and in the British North American colonies for over 200 years.

Hence the importance of fact-checking. Fake news didn't originate with the internet. Or even with the printing press.

By 1822, hundreds of tomato recipes appeared in local periodicals and newspapers, but fears and rumors of the plant’s potential poison lingered.

Again, a testament to the problem of anchoring bias, where the first piece of information you learn about something sticks with you in spite of later corrections. Still, if that first piece of information is "that shit'll kill ya," this can be somewhat forgivable.

Today, tomatoes are consumed around the world in countless varieties: heirlooms, romas, cherry tomatoes—to name a few. More than one and a half billion tons of tomatoes are produced commercially every year. In 2009, the United States alone produced 3.32 billion pounds of fresh-market tomatoes. But some of the plant’s night-shady past seems to have followed the tomato in pop culture. In the 1978 musical drama/ comedy “Attack of the Killer Tomatoes,” giant red blobs of the fruit terrorize the country. “The nation is in chaos. Can nothing stop this tomato onslaught?”

Awww. I was hoping that the article wouldn't mention this seminal work of cinema, so I could write this entry about it. Oh, well, I guess I'll just keep it in the title.

There's more to the article than the bits I riffed off of here; it's worth reading for the description of the dreaded Green Tomato Worm (which is not, in fact, a worm that infests green tomatoes, but a green worm that infests all sorts of tomatoes -- one of the many ambiguities possible with the English language, which is what makes it so much fun).

But mostly I'm glad this article came up in my random roll today, because it's science, history, food, cinema and the potential for comedy all rolled into one, like a burrito. A burrito with chopped tomatoes. Okay. Now I'm hungry. Tomorrow's entry will almost certainly be darker, for reasons that should become apparent.

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