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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 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 4 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.

August 26, 2021 at 12:02am

#1016147

Mad Scramble

This isn't a source I'd normally link to, but Someone is Wrong on the Internet and I cannot let it stand.

There’s Only One Right Way to Cook Scrambled Eggs
It’s also the easiest.

Now, I'm all for "the easiest." As you know if you've been following along. Anything that saves me extra work is a good thing -- except when it comes to the taste of food or drink. For example, you'd think I'd be putting oatmeal into the microwave: it saves time and dishes. But microwave oatmeal tastes to me like paper pulp, whereas stove-cooked oatmeal tastes like marginally less disgusting paper pulp. Therefore, I put up with taking a little extra time and having to wash a pot and stirring spoon afterwards.

So it was with some interest that I came across this "you're doing it wrong" article, only to find my hopes dashed like the eggs I use for scrambling.

I’ve eaten at least two eggs almost every day of my adult life. Eggs are good for you. They’re high in protein, low in calories, and contain a host of essential vitamins and minerals.

This guy must not have been an adult very long. In my lifetime, eggs have gone from "good for you" to "bad for you" to "good for you" to "bad for you" to "just the whites" to "bad for you" and finally back to "good for you," where they seem to be stuck like they do when you run out of treats for the neighborhood vandals on Halloween.

I’ve eaten eggs many different ways, but I usually eat them scrambled.

Okay, I'm with you so far.

Poached eggs are wonderful, but making them undercaffeinated is a terrible way to wake up.

Fair.

Over-easy eggs are also good, but sometimes you will break a yolk and then ultimately you will make scrambled eggs, which is also why I usually eat scrambled eggs.

Sometimes I think I'm the only non-vegan in the universe who actively hates runny yolks. I do not like them on my bread; I do not like them in my head. Fuck runny yolks.

What I’ve come to understand is the there’s only one right way to cook scrambled eggs.

Thus begins the mansplaining. To be fair, though, the target audience is other men, so... whatever.

Step 1: Use good eggs.

This is important.

You know, ideally, sure. In reality? I wake up with a hankering for scrambled eggs, and I use whatever's in the fridge, which has usually been sitting there in the carton for a couple of weeks. So, no: I'm not buying a new dozen eggs every time I get groceries delivered, and throwing out the still-viable old ones, just because it's better to work with, and eat, fresher eggs.

Farm-fresh (or at least farmers’-market-fresh) eggs taste, well, like eggs. They are bright, clean, and substantive.

And I am absolutely not going to go out of my way to buy artisanal, free-range, hipster, locally sourced cackleberries. I don't have a car, and besides, then I'd have to deal with *shudder* people.

Step 2: Use great butter.

This is very important.

What's not important, then?

See above; same as regards to butter. Whatever. Is. Available. From. Delivery.

Higher-end butter (and I’m only talking a buck or so more here, cost-wise) tastes rich and creamy. It possesses a certain heartiness.

I should write for Men's Health. Apparently if I did, I'd earn tons of disposable income.

Step 3: Turn the heat down.

Most diners don’t cook eggs well. That’s because the short-order cooks are blasting all the food over a high-heat griddle. That means you get your eggs fast, but they’re often as dry as the paper placemat.

Okay, I'm with him on this one. Lower heat cooks the eggs just fine; it just takes a little longer and you don't have as much risk of overcooking.

Step 4: Don’t beat it.

HERE WE GO INTERNET.

DAMN RIGHT.

I don’t beat my eggs separately in a bowl. It’s a waste of time. It’s a waste of a bowl and a whisk and minute of your morning.

While I appreciate the laziness -- I really do -- I prefer beaten eggs. Period.

I add a thumb-sized amount of butter to a small non-stick pan, allow it to melt, swirling to coat, and then just crack the eggs into the pan.

Then I don’t do anything until I begin to see the egg whites begin to cook. At this point, I take my spatula (I prefer rubber, but whatever) and stir the eggs.

Something I hate almost as much as runny yolks is flecks of white in my scrambled eggs. Okay, this guy's fine with it. To each their own. But it's not objectively "better." It's a matter of taste, and visuals. I want my scrambled eggs to be a uniform, pale yellow color, not marbled like the lobby of the goddamn Ritz-Carlton.

Besides, beating the eggs first introduces some air, which again, I prefer.

Step 5: Don’t add anything.

People do weird things to scrambled eggs as they cook. They add cream. They add salt. For some reason they add pepper or chives. Stop all that.

It's not scrambled eggs unless you add some black pepper during (or before) cooking. Well... for me, anyway.

Chives are a garnish added after the fact, and besides, who the hell can keep fresh chives around? For me, they go bad the next day, just like any fresh green thing I buy, and dried chives are tasteless.

After you’re done scrambling the eggs, you can candy them in ketchup or smother them in hot sauce for all I care, but during the cooking process don’t fuss with them.

Damn right hot sauce. I prefer Marie Sharp's, imported from Belize. Because I, too, can be a pretentious asshole.

Step 6: Power down and plate.

The last minute of egg scrambling, I’d argue, is the most important.

Oh, look, another important thing.

How you will know you’ve entered the last minute: Your eggs will begin to resemble the consistency of folded satin. There will be the appearance of solids forming, yet the eggs will still look smooth.

What the hell even does this mean?

This isn't "the only way to cook scrambled eggs." This is "One guy's preferred method of cooking scrambled eggs." He's not doing it wrong. I'm not doing it wrong. You might be doing it wrong, but the only way to do it right is to a) know the way you like them and b) figure out how to get to (a).

These days, I don't even bother with scrambling them. I wilt some (prepackaged because I'm lazy) baby spinach in butter or olive oil along with a bit of salt and some black pepper, pour two beaten eggs on, pop a top on the pan, and leave them alone until the top's no longer liquid so it doesn't all run off when I flip. I flip once, and done, and I don't even try to do the fancy pan-flip, because who am I trying to impress? I suppose that's technically a frittata. I just like saying frittata. Frit. Tata. Say it. Revel in it.

Or I could, I suppose, get a supply of frozen breakfast wraps. Yeah, that's the ticket. Of course I'd have to take the time to cook them in the toaster oven, though, because the frozen wrap does not exist that, when microwaved, doesn't have pockets of ice in it. So frittatas it is.

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