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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February 26, 2021 at 12:03am

#1005286

Forbidden fruit

Ever wonder about grapefruit?

Grapefruit Is One of the Weirdest Fruits on the Planet
From its name, to its hazy origins, to its drug interactions, there’s a lot going on beneath that thick rind.

You know, there are quite a few weird fruits on the planet. Durian comes to mind. And wtf is up with breadfruit? But yeah, okay, maybe grapefruit is a bit weird, especially since grapes are already fruits and they have nothing to do with citrus so what's with the name?

Right from the moment of its discovery, the grapefruit has been a true oddball. Its journey started in a place where it didn’t belong, and ended up in a lab in a place where it doesn’t grow. Hell, even the name doesn’t make any sense.

See?

The current theory is that somewhere around five or six million years ago, one parent of all citrus varieties splintered into separate species, probably due to some change in climate. Three citrus fruits spread widely: the citron, the pomelo, and the mandarin.

You don't see pomelo much around here. I have a vague memory of eating one, long ago, in a foreign land. At the time, I thought it was a cross between an orange and a grapefruit. I guess that was backwards.

With the exception of those weirdos like the finger lime, all other citrus fruits are derived from natural and, before long, artificial crossbreeding, and then crossbreeding the crossbreeds, and so on, of those three fruits. Mix certain pomelos and certain mandarins and you get a sour orange. Cross that sour orange with a citron and you get a lemon. It’s a little bit like blending and reblending primary colors. Grapefruit is a mix between the pomelo—a base fruit—and a sweet orange, which itself is a hybrid of pomelo and mandarin.

Yeah, I know. I got lost too.

Speaking of all these names, let’s discuss the word “grapefruit.” It’s commonly stated that the word comes from the fact that grapefruits grow in bunches, like grapes. There’s a pretty decent chance that this isn’t true. In 1664, a Dutch physician named Wouter Schouden visited Barbados and described the citrus he sampled there as “tasting like unripe grapes.” In 1814, John Lunan, a British plantation and slave owner from Jamaica, reported that this fruit was named “on account of its resemblance in flavour to the grape.”

Yeah... no.

This is largely guesswork, almost all of it, because citrus is a delightfully chaotic category of fruit. It hybridizes so easily that there are undoubtedly thousands, maybe more, separate varieties of citrus in the wild and in cultivation.

Seriously, though, the vast variety of citrus and its ease of modification is pretty fascinating.

The article goes on to describe how grapefruit, and other citrus, led to Florida becoming Florida, so there's another reason for me to hate grapefruit.

It also talks about the very interesting discovery that grapefruit completely fucks with some medications.

Now, I've said this before but I'll say it again: I've never liked grapefruit. I mean, I never really hated it; if it's there I'll eat it but I never sought it out, or deliberately obtained grapefruit juice to drink. It was just something that was there. That is, until I started taking a statin, at which point I got really intense cravings for grapefruit.

The surest way to get me to want something is to tell me I can't have it. I mean, it's possible that if someone told me "you can't eat eggplant or your blood pressure will go through the roof," I'd want to go out and buy bushels of eggplant.

Possible, but I doubt it. At least I always acknowledged that grapefruit was edible.

Anyway, I'm not going to quote the circuitous part of the article that goes into the discovery of grapefruit interactions, but basically, it can have the effect of making us metabolize more of certain medicines than expected.

I know a guy who takes advantage of this. He's poor and has shit insurance, so he stretches out his statin supply by taking 1/2 the recommended dose and munching on grapefruit.

Pretty sure that's not recommended.

“There are a fair number of drugs that have the potential to produce very serious side effects,” says Bailey. “Kidney failure, cardiac arrhythmia that’s life-threatening, gastrointestinal bleeding, respiratory depression.” A cardiac arrhythmia messes with how the heart pumps, and if it stops pumping, the mortality rate is about 20 percent. It’s hard to tell from the statistics, but it seems all but certain that people have died from eating grapefruit.

And see, I'd rather die from eating something that I actually like.

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