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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October 10, 2021 at 12:04am

#1019041

There's No Ham in Hamburger

I like to see journalism that covers the most important issues of our day, things of great import with far-reaching repercussions.

The Best Order to Stack Your Burger Toppings

Yes, I follow a site called The Art of Manliness. Shut up.

The big keys to cooking a great burger center on using the right meat and grilling technique.

There are other acceptable ways to cook a hamburger patty, but we all know that grilled is superior.

But once you’ve got those fundamentals in place, the architecture of your burger can also enhance the taste and overall eating experience.

I'd also add that choice of toppings is essential. Take the tomato, for example. There are approximately 158,343 varieties of tomatoes in existence. You can rule some of them out right away: cherry, Roma, etc. Too small. You'd think "beefsteak" tomatoes would be best because they're enormous and one slice can just about cover the whole patty, but beefsteak tomatoes taste like cardboard. It's a variety sold not just for its size, but because it stacks without mushing, travels well and ripens in transit (or something like that). But no - one thing that spoiled me growing up on a farm was access to fresh vegetables such as vine-ripened tomatoes. As long as a vegetable didn't need much preparation, my mom couldn't destroy it by cooking. Size is a bit less important than flavor, here. I know not everyone has access to fresh, off-the-vine tomatoes, so my privilege is showing, but dammit, what's the point of privilege if you can't eat decent food?

Personally, I can take tomatoes or leave them. I don't understand the over-the-top love for those squirty things. But if I'm going to eat a tomato, I get to be picky about it. (In addition, don't refrigerate them if you can avoid it; they become mealy.)

Also, never, ever put iceberg lettuce on a burger. Iceberg, fresh or not, is like 98% water, 1.5% sadness, and 0.5% apathy. Pretty much any other variety of lettuce is superior.

My take on other burger toppings is similar. The article covers the classics. Of course, I have Opinions, which might as well be Facts in this case.

Plenty of people structure their burger willy-nilly, and really, there’s no wrong way to do it. But there are ways that make it more or less likely that you’ll have issues with things like soggy buns, suboptimal flavor melding, and, most vexing of all, escaping toppings; it’s not fun to have those toppings squirting and falling out as you try to get your delicious meat sandwich in your mouth.

No, the structure is certainly a matter of personal preference, though the article goes on to make some good suggestions for optimal construction. But I just gotta say: nothing pisses me off more when I'm eating something than if it makes a mess. If it's a food that's supposed to be eaten in the hand -- a sandwich, a taco, a sub/hoagie/grinder/whatever, fried chicken; pizza, anything that's meant to be shoved into your gaping maw by your grubby paws instead of utensils -- I utterly despise having shit falling out the other side, getting on my clothes or the table. It's okay if a little gets on my hands, but licking it off is an absolute no-no. I've lost friends over this. They'll suck their fingers and I'll be like "Ew, go away." We invented napkins, shirtsleeves, and pantlegs for a reason.

You'll note I didn't put hot dogs in the above examples. That's because hot dogs are already included. No, they're not sandwiches. They're tacos. Fight me.

The article builds a burger from bottom to top. I'll address them in order.

Bottom bun.

Choice of bun is also of great importance. To me, the purpose of eating anything is the bread. If it's not a bread I wouldn't eat alone, I don't want to stack shit on it. You know what I like to use for a bun? An English Muffin. There, I said it. Not the flimsy crap ones with the Thomas' label, but a nice thick crispy-outside-chewy-inside toasty bready thing. Kaiser rolls are also acceptable, as are many other types of real bread. What's not acceptable? Wonder bread and its ilk.

First condiment of choice. Distribute your condiments (mustard, ketchup, mayo, etc.) between the top and bottom buns.

Mayonnaise does not belong on a burger. Save that shit for BLTs, tuna salad, potato salad, and for luring Midwesterners into your spice den.

Also, don't cheap out on the condiments. French's Yellow Mustard says "I don't care about actual flavor; I just bought whatever costs the least."

Lettuce. With its placement atop the bun, the lettuce acts as a shield that blocks the patty’s juices from soaking into it and making it soggy.

Again, this has to be the right kind of lettuce (anything but Iceberg). And dammit, don't use shredded lettuce of any variety. That subverts half the purpose of putting lettuce on in the first place, which is to keep the bun from getting soggy.

Tomato. Slippery veggies like lettuce and tomato are best placed beneath the burger and are less likely to slip when used as foundational pieces rather than sitting on top of the patty.

You know what contributes more to general burger sogginess than the patty? The damn tomato. Often I'd leave it off entirely, if I can't exercise my snobbery, detailed above, about this vegetable. I can't tell you how many times I've been in a restaurant and ordered a burger, only to find the tomato is half-green, crispy, and, worst sin of all, contains part of the stem attachment.

Burger patty with melted cheese.

There's a link at the original article to the "best" choice for patty meat. Personally, I prefer the more lean ground beef. If you're into other meats besides beef, that's fine. And I'm going to piss off my fellow carnivores here by saying: Veggie patties are okay, too. I haven't tried the new almost-meat varieties yet; traditional veggie patties are very dry, making condiments more important. I'd even forgive mayo on such a burger.

And you'd better believe I have opinions about the cheese. I did a whole blog entry a week or so ago on the glories of cheese. There's a lot of room for personal taste here, sure, but that "pasteurized process cheese food" plastic bullshit is never acceptable on a burger.

Onions.

Ideally, red onions. White onions are okay, too. NOT yellow onions; those are for cooking. Which reminds me, it's almost time for me to make my French onion soup again. And my chili. I guess there are some good things about cooler weather, but this entry is most likely the last time I'll admit that for a while.

Pickles.

Yes, pickles are an essential part of any burger. Hell, if I'm feeling lazy, that will be the only topping I put on besides cheese. Lettuce has to be washed before it's used; tomatoes, washed and sliced; and every time I slice an onion I'm convinced I'm about to chop off a finger. But pickles? Pull out of jar. Slice lengthwise so there's less chance of topping displacement. Also I can't keep a vegetable in my fridge for more than a couple of days before it becomes a biology class project, but the whole raison d'être for pickles is they keep a long time.

That said, I'm a snob about pickles, too. Dill is fine. Half-sours are superior, if you can get them. Bread & Butter pickles can fuck right off, as can any labeled "sweet." Those have their place, but that place is not on my burger.

Second condiment of choice.

Since mayo is absolutely unacceptable, this will be ketchup or mustard or, as is my personal preference, something that bites back like, say, ghost pepper sauce or something with habaneros.

Top bun.

Now I have an idea for a Tom Cruise movie parody.

So that's the factsmy personal opinion about burgers. I'm sure you have different opinions. Therefore, we'll do another contest.

Merit Badge Mini-Contest!

In the comments below, describe your perfect burger. It can be one you've had or, as in my case, a Platonic ideal that I've never quite achieved (though I've come close). It doesn't have to have the "classic" toppings; I also like, for example, mushroom-swiss burgers. Nor does it have to contain dead animal. My above rantings (which really should be taken humorously) aside, I know that tastes vary, so even if it's something I don't like, that's fine (though admittedly, you're going to have to be pretty convincing if you include mayonnaise). As always, the deadline is midnight at the end of today, Sunday October 10, according to the WDC clock. The best description (again, not necessarily one that I'd like) will earn its chef a Merit Badge on Monday.

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