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.

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.

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

1 Like

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

November 16, 2022 at 12:01am

#1040706

Time to Cook

Yeah... I can relate.

The 30-Minute Meal Myth

Of course, ordering a pizza takes less than 30 minutes. Sure, you usually gotta wait longer than that for it to show up—Domino's nixed the whole "30 minutes or less" thing years ago, and they suck anyway—but at least you can be playing video games while you wait.

We’ve all fallen for the trap before. Wooed by the promise of pan-seared chicken thighs in 30 minutes, only to be defeated and left overanalyzing what went wrong more than an hour later. Or worse, we’ve thrown some onions in a pan to caramelize while we’re searing a batch of burgers, only to find ourselves still stirring the onions dejectedly, 45 minutes later.

It's not just recipes, either. "Minute" rice can take way longer than a minute to cook.

It’s right there, staring at me. Cook time: 30 minutes. But a closer look at the ingredients says otherwise. Five garlic cloves, minced. One stalk of celery, thinly sliced on the bias. Two carrots, peeled and chopped. One yellow onion, finely diced. There go 15 minutes already (on a good day, with a sharp knife, and no distractions), which doesn’t even account for the five minutes needed to compose myself after tearfully hacking at an onion. And that’s only half the battle, if we’re counting the unglamorous process of washing and thoroughly drying all of those vegetables.

Not to mention the half hour or so you spend cleaning up what your roommate left in the sink and on the stovetop.

Look, I'm a big fan of mise-en-place, and was even before I started seriously learning French. Get all that measuring and chopping crap out of the way before you start cooking and you're not stuck watching your pot burn while chopping the onions in the middle of it all. There will be at least one onion that's started to go bad, too, so you always use more onions than you think.

I have managed to keep onion juice from messing with my eyes, though, so there's that.

But the problem with mise-en-place is you're not multitasking, so it usually takes more time to cook something if you're careful about getting everything all set before you fire up the stove.

Recently I fell into a similar trap after being convinced by a trusted blog that 35 minutes was all I would need to make mapo tofu in my Brooklyn kitchen.

Gotta get that humblebrag in. At least in Brooklyn, if you suddenly find you're out of ingredients, it's a much quicker trip to get more than in most parts of the world.

After pulverizing Sichuan peppercorns with a mortar and pestle, peeling and mincing a three-inch knob of ginger, finely chopping half a head of garlic, and rummaging through my dish rack to get enough small bowls and spoons to premeasure the rest of the ingredients, I’d already blown past the 20-minute mark, and I hadn’t even turned on the stove.

The peppercorn thing is way too much work. Ginger is way too much work, too, but it's worth it. And don't get me started again on peeling garlic.

Beneath a fish taco recipe advertised as a “fast dinner for hungry, busy people” in the New York Times, a comment reads, “It’s unbelievably condescending to claim this meal takes 30 minutes. It took me 15 minutes just to make the salsa, 7 for the mayo, 10 to warm all the tortillas, and a full 30 to fry all the fish in batches…. Great recipe, horrifically underestimated execution time, especially for those with kids running around.”

If it's taking you 10 minutes to warm up tortillas, either you're feeding the Mexican army, or you're doing something very, very wrong.

The conditions we’re under have their own matrices of variables. “Part of it is that recipes don’t account for skill levels—such as how fast you chop or mince and the equipment you have at your disposal,” says Kelly Chan, a Queens-based nonprofit analyst who’s often folding dumplings or prepping Cantonese-style stir-fries. Recipes are written with the presumption that all home cooks have speedy, chef-like knife skills to whiz through a mountain of shallots and tomatoes, or that they know how to butterfly a chicken without pausing, washing their hands, and looking up a YouTube tutorial. Even the Instant Pot—widely adored among home cooks for its shortcuts to complex 20-minute pho broths or five-minute steel-cut oats—still needs time to preheat and depressurize, effectively tripling the cooking time in some cases. (But of course no one tells you that, because it’s called an Instant Pot for a reason.)

I've never used an Instant Pot, but my gut told me the name was an exaggeration. Not just that, though, but also the work involved in cleaning it keeps me from buying one (my housemate has one, but rarely uses it, and I'm concerned I'll muck it up). Every time I consider a new kitchen gadget, I mentally figure out how much work cleaning it will be, and usually don't bother. One exception is a blender; those are usually worth the work to clean.

Real cooking proficiency isn’t about whipping things up without a recipe—it’s about reading between the lines of that recipe and knowing when an hour means two hours.

I usually mentally double a recipe's stated cooking time, and it still often runs longer than that. One time, I was trying to make latkes. I knew going in that it would be labor-intensive; that's just the way it goes. What I didn't account for, though, was that the damn things took three times as long to cook as I expected. To be fair, this doesn't always happen; one should use russet potatoes for latkes, and I had to get some other kind because the store was out of russets (this was around Hanukkah a couple years back; I guess everyone else was making latkes, too. Everyone's Jewish for latkes.)

My biggest gripe about cooking for myself, which is the usual case, is that I generally think that a dish shouldn't take longer to cook than it does to eat. Sometimes I do it anyway, for practice. But after laboring over a hot stove for two hours and finishing the resulting meal in less than five minutes, I'm left with the distinct impression that I've wasted my time.

Maybe I should buy more frozen dinners.

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