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.

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.

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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.

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').

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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

October 1, 2021 at 12:03am

#1018454

Praise Cheeses!

Let's kick off October with an entry for "Journalistic Intentions"

[18+]

Cheese

Back in the noughties, I visited the UK for a couple of weeks, and spoke to a lot of nice British folks (and some who weren't so nice, but not for quite as long). "Oh, you're American," they said as soon as I uttered a single word. "You must like Bud Light."

Well, no, I hate that watered-down fake excuse for beer. Even then, I was a craft beer snob, and I told them so. "Actually, there are thousands of local and regional craft breweries in the US, and I've made it my life's mission to try them all."

It's similar there, only they called it "Real Ale." Whatever; I'm used to us having different words for the same concept.

"Well, what about wine? Surely you get all your wine from France."

"Actually, while French wine is quite good, we have thousands of local and regional vineyards producing spoiled grape juice. In fact, I live very close to the place that perfected the science of winemaking in less-than-ideal climate conditions."

I got, "Wow, the US is more advanced than I thought." (I decided not to contradict them on this point.) "Well, what about cheese?"

"...okay, you got me there."

The UK produces and exports a wide variety of cheeses, mostly named after their region. The most famous and widespread (pun intended) of these is, of course, Cheddar. While there are certainly some small cheesemakers in the US, most of the crap you find in stores is industrially prepared, quality-controlled, and distributed nationally, and as far as I know, we haven't innovated our own cheese styles like the British (or French or Swiss or anyone else) has. One exception is Monterey Jack, .which, while American in origin, is more Mexican in ancestry, as it came from that part of California. Most of the real cheese produced in the US seems to be copies of European styles: Gruyere, Brie, Gouda, Stilton, etc.

A lot of the cheese sold here isn't "cheese" in anything but the most legal, FDA-approved sense, in that it passed near a cow on its way from the factory to the store. The worst offender is the plastic "pasteurized process cheese food" usually marketed as American cheese; or, as I like to put it, the dairy counterpart to Bud Light. Of course it's called American: it's bland, fake, and uncultured.

While I admit to a) liking that stuff as a kid and b) buying mass-produced real cheese from the store because it's cheaper, I'm just as much of a cheese snob as I am a beer and wine snob. Very little pleases me more, where food is concerned, than walking into a new-to-me grocery store and discovering that they have a wide variety of domestic and imported cheeses.

And I like all of it.

Some more than others, of course. Soft cheese, hard cheese, cottage cheese, moldy cheese, whatever, doesn't matter. There's also an art to pairing different cheeses with different wines or beers. As with all art, I suck at it, but I can appreciate others' efforts.

One reason I will never be vegan is, in fact, cheese. Not meat -- I could suffer through the rest of my life without meat if I had to, but you will have to pry my cheese from my cold, dead, cholesterol-soaked hands. There are simply no good vegan substitutes for cheese, though some of the meatless "meats" are quite tasty.

Speaking of the US riffing off the UK, and cheese, I found this on the internet, so I don't know who to attribute it to. But it's relevant:

Sweet dreams are made of cheese
Who am I to dis a Brie
I cheddar the world and a Feta cheese
Everybody's looking for Stilton

Though someone went on to do a complete parody of the song

, and it's hilarious, especially because I can't stand the original.

In closing, you know how YouTube generates recommended videos whenever you're there? Well, when I watched the video that prompted this entry (you'll have to click on the contest link above to get a link to it), this was the top recommendation. Not really relevant to cheese (the history of both cheese and domestic cats seems to be traced the Fertile Crescent at about the same time, though this is probably a coincidence), but YouTube has me pegged: it's definitely relevant to my interests:

7 Like • Comment • View 6 comments

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