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 23, 2021 at 12:02am

#1005046

United Sandwiches of America

Well, it looks like I have a new mission in life. In addition to the beer odyssey.

The Best Sandwich in Every State

The mission is to visit every state and verify -- or disprove -- this list.

"All of humankind has one thing in common—the sandwich," renowned late-aughts philosopher Liz Lemon once theorized, on NBC's 30 Rock. "I believe that all anyone really wants in this life is to sit in peace and eat a sandwich."

Late-noughties. There's no excuse to call the 2000-2009 decade anything else. (Spare me the pedantry of pointing out it's "really" 2001-2010.)

Roughly as old as the country and invented by the Earl of Sandwich, an Englishman who never seemed to have time for a proper sit-down meal, Americans have spent the entirety of our nation's existence seeking to perfect the humble art form.

You know, that "Earl of Sandwich" origin story never quite sat well with me, like the aftereffects of... well, of a sandwich that's maybe a bit too greasy.

https://en.wikipedia.org/wiki/Sandwich

The sandwich is named after its supposed inventor, John Montagu, 4th Earl of Sandwich. The Wall Street Journal has described it as Britain's "biggest contribution to gastronomy."

If you follow the links from that Wikipedia page, you'll find that Montagu, Earl of Sandwich, didn't invent the sandwich any more than Amerigo Vespucci "discovered" America, and yet they became their respective namesakes. So okay, fine, the concept existed before they named it that.

Anyway, back to the original article.

There were strict parameters; no burgers, no hot dogs, no burritos, no tacos, and in nearly all cases, no barbecue. Sandwiches or not sandwiches, they can go ahead and get their own lists.

Want to start an argument? Ask if a hot dog is a sandwich.

I mean, sure, it's meat and stuff eaten with bread, but technically it's not two pieces of bread, but a single hinged bun. Consequently, it's a taco. I have spoken.

Now, obviously I'm not going to comment on all of the examples here. For starters, I haven't eaten any of the examples. But I do have some things to say -- obviously.

California

If anyone asks you what the 1970s were like in Los Angeles, drag them down—immediately, if not sooner—to Langer's Deli, the best Jewish deli in America, for the pastrami.

That's a bold statement right there. Now, granted, I haven't been to Langer's, but for some reason pastrami sandwiches are quite popular throughout the West, and I've had a few -- and not a single one of them can compare to the ones you get in New York City (or even one I've had right here in my Virginia town). However, I'm a reasonable person, and will withhold judgment until I get to L.A.

At least they're better than most of the South, which somehow came to the mistaken conclusion that it's okay to serve cold pastrami. It is not.

Kentucky

We like to think of the hot brown as the beginnings of a fine turkey club: roast bird, strips of bacon, and slices of tomato on toast. Plot twist—the whole thing is then flooded with rich Mornay sauce before hitting the broiler, emerging a delicious mess that requires a knife and fork to consume.

I have no doubt that it is delicious, but this is not a sandwich. Yes, I know that "open-face sandwiches" are supposedly a thing, but if it's a piece of bread with toppings on it, it's not a sandwich; it's a pizza. Regardless of toppings.

Massachusetts

Surely there are lobster rolls in a coastal state bookended by Maine and Connecticut, but we're too busy filling up on clam rolls, which are the first meal we think of when we think Massachusetts, or at least the very large amount of the state located by the ocean.

I have nothing to say about this creation that I haven't tried yet, but I'm including this to point out that I went to a BBQ place in central Massachusetts once that billed itself as having "The Best BBQ in the US." Being from the South, I had to go in and dispute that claim. Well, I did -- but I have to admit, it didn't suck.

Nebraska

...that other delightful, if less widely-renowned, Nebraska invention, the deep-fried grilled cheese sandwich.

Go. Go look at the picture of it at the link. I'm getting a heart attack just looking at it.

Pennsylvania

A serious jostling in the mosh pit that is South Philly's perennially cramped John's Roast Pork rides high atop our list of post-pandemic musts, partly just to feel something after our terrible year of No Touching, but also for a sandwich, a roast pork sandwich please, the one any right-minded Philadelphian can tell you is the one you go looking for once you're ready to dine sober, in the sunlight, like a whole adult.

I'm including this one so my friends from Pennsylvania can go find the article's authors and shove some Philly cheesesteak down their gullets.

Virginia

In a perfect world, the I-81 struggle through America's Secretly Biggest State would have been long ago ameliorated by modern conveniences including a third lane in each direction; in the meantime, better to think of this essential leg of the Northeast to Deep South fast route in terms of the many small detours one can make in order to feel human again, after yet another hour or staring at the rear end of the same tractor trailer.

Virginia is not a state to travel through. It is a state to travel to. Or preferably to be from.

I haven't tried the sandwiches mentioned here but you'll note that unlike most of the other states' offerings, there are two choices. Now I have to try them, but I seriously doubt they'll be better than the pastrami and swiss with spicy mustard and onion on a sub roll that's served at a lunch counter here in my town.

So... road trip, anyone? As long as I also get to try the nearby breweries.

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