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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 10 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

1 Like

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').

1 Like

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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

June 15, 2022 at 12:02am

#1033837

Egg Sellent

After yesterday's entry, I'm sure we could all use a little less depressing fare. I have eggsactly the thing.

How to Make a Perfect Over-Hard Fried Egg

As the site is called Art of Manliness, one might eggspect that the whole page would be "make your wife do it." You sexist pig. You need to know how to do it yourself because she will divorce you with that attitude. No yolk.

There are a bunch of different ways to cook eggs: hard-boiled, poached, scrambled...

I did a thing on hard-boiled just a couple of weeks ago: "Egg Zactly"

...heck, you can even make them in the microwave.

Oh yes, please. Make sure you get it on video. (Seriously, though, it can be done, just not whole eggs.)

When it comes to making an egg sandwich or putting an egg on a burger (and a burger should seldom be without one)...

Lies.

...there’s no question as to which method is best: fried. All the way.

You know, one thing that struck me while comparing this article with the one on hard-boiled eggs: this one makes pronouncements; that hard-boiled egg one did science.

You can fry an egg over-easy, over-medium, or over-hard, depending on how runny you like the yolks. Folks who don’t like any runniness like their eggs cooked in the latter style.

That would be me. There's a lot of things I don't understand when it comes to peoples' tastes, and near the top of that list is the runny-egg thing. Fortunately, I don't have to understand it; I just have to accept it, like I have to accept that some people like anchovies on their pizza or bacon that droops after it's cooked. Just as long as they understand they're wrong.

The full deliciousness of an over-hard fried egg lies in the maintenance of its original architectural composition: a wholly intact yolk, sitting in a wholly intact white.

Eggineering. Not architecture. (You pronounce that "egg-gin-ear-ing.")

Yet getting both parts of the egg to cook evenly, without breaking the yolk, can prove challenging.

The rest of the article goes into the actual method, which, to be clear, is very similar to what I do anyway. But yes, sometimes when I flip the egg I end up with a suboptimal result. I attribute that to not doing it very often, so I'm out of practice. And it hardly matters, since I'm just cooking for myself anyway; still, I have a perfectionist streak when it comes to cooking. And writing. And, okay, pretty much everything.

But that's not the only challenge I face in the kitchen. I have yet to find solutions to these issues:

Cling wrap. I've pretty much mastered the art of not letting the end fall back onto the roll, thus making it utterly impossible to restart it. But two things elude me. First, being able to get a clean cut after I roll some out; inevitably, it'll tear someplace other than the tear strip thing on the box. Or, usually, start at the right place and then go off on a diagonal somewhere. Second -- and most frustrating -- is that I have yet to be able to pull out any length of cling wrap without the stupid stuff sticking to itself; and then, of course, it never actually clings to itself, or to the container, when I use it for its intended purpose. This is incredibly frustrating and has led me to invent creative curses. And one of these days I'm going to use this in a story: "Her name was Sarah, but we called her Saran, because she was clingy as hell except when you wanted her to be."

Along those same lines, aluminum foil. How the hell does anyone get it to stay smooth? Or even fold nicely? I go to wrap leftover pizza slices in it and it looks like a damn moonscape; that is, if I don't tear it during the attempt.

I'm lazy, so I keep a stash of frozen egg-meat-cheese sandwiches to nuke for breakfast. The instructions are usually something like: Wrap sandwich loosely in paper towel. Microwave on low for 90 seconds to thaw. Flip, nuke on high for one minute to cook. So I do exactly what it says, and half the putain fromage has melted onto the goddamn paper towel. Now, for a while there, I was also having trouble with the wrapping part. If I just folded the paper towel over the sandwich, it would flower open in the microwave and ruin the taste. I finally started to do an envelope-style folding thing (an envelope was what we used to send handwritten letters through the mail when there were handwritten letters and mail). By locking one corner under the other three, I managed to keep the package together -- but this didn't solve the waste-of-lousy-cheese thing.

Ever buy one of those nonstick cookie sheet pan things? Were you able to use that expensive piece of crap more than once? I can't. First thing I bake on it stains it permanently. I'm not talking about a small discoloration here, but a complete ruination of the supposedly nonstick surface. Maybe I'm just buying the wrong kind of cookie sheet, but I have yet to find one that is more than single-use. Because of sunk costs, I usually end up lining the damn thing with crinkly-ass aluminum foil, and then spraying the fuck out of said foil with Pam.

Compared to those frustrations, getting a fried egg wrong is bush-league stuff. It's enough to make me want to overuse Uber Eats.

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