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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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.

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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.

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').

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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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

August 24, 2022 at 12:02am

#1036871

Peeling Out

Today, some hard-hitting journalism sure to be eligible for a Pullet Surprise.

Is Garlic Getting Easier to Peel?
To solve the mystery, I had to talk to horticulturalists, farmers, chefs—and also my local grocery store.

Mmmm... garlic chicken...

Where was I? Oh yeah. Most of the time, the answer to any question posed in a headline is "No."

For most of my adult life, I was a garlic-phobe. Not because I didn’t like the flavor! No, like basically every human on earth, I adore garlic...

I've known people who didn't like garlic. I avoid them, figuring they're vampires.

I loved eating garlic. What I hated, for years, was peeling garlic.

If only you could have relaxed your culinary snobbery long enough to purchase pre-peeled garlic. Or use (gasp) garlic powder. It has its place, you know.

What a drag it was! Let’s say your recipe calls for three cloves of garlic. Your onions are fizzing, your pasta is bubbling, and you’re cursing, trying to separate the garlic’s skin from the cloves within.

You know, I don't consider myself a gourmet chef, though I get by. And I've ranted before, in the Comedy newsletter and probably here too, about the frustration of garlic-peeling. But there's a concept called mise-en-place, which I knew about even before I started learning French. Literally "put in place" (the "place" is pronounced differently, of course), it means prepping most of your ingredients before starting cooking, so you don't find yourself struggling with peeling garlic, chopping onion, or whatever at critical stages in your cooking.

It's not always necessary. If you're broiling chicken or whatever, you can do some chopping while it's cooking. But having the garlic peeled and minced before you even turn the stove on is, in my estimation, pretty basic.

Like juicing a lemon or opening a stuck jar, peeling garlic is one of those kitchen tasks so pesky that gadget companies are forever trying to solve it for you. I’ve tried the shakers. I’ve tried the silicon rollers. I’ve tried cutting off the stem end and rolling the clove between my hands.

Cut off the stem end, twist the cloves a bit to loosen the skin (this also releases some flavor), and rub them together between your hands while standing over a trash can to catch the paper. You do run the risk of dropping the garlic, so don't do that. Notice I said "cloves." No matter what a recipe says, a single clove is never enough.

NaNoNette

recommends squeezing them under the flat of your vegetable knife, and that can work too, but it requires more tools than just using your fingers.

Also, juicing a lemon? Come on. The only trick to that is to do it through a sieve so the seeds don't squirt out onto your shrimp or whatever.

But over the past three years or so, something strange has happened to the garlic I buy at the grocery store. It’s become so much easier to peel!

The more you do something, the easier it gets.

How did my garlic transform from sticky nightmare to user-friendly flavor dispenser? Have America’s garlic breeders suddenly focused on peelability as a saleable trait? Has competition from pre-peeled garlic somehow forced a change in the garlic farming world? I had to know the answer, so I called everyone I could think of who might know anything about garlic.

So, not vampires.

“I don’t know that anybody’s measured that, peelability,” said Barbara Hellier, a horticultural crops curator with the United States Department of Agriculture in Pullman, Washington. Like a home chef with a particularly tough clove, she wrestled to unwrap the subject:...

Stretch metaphor.

I asked if anyone was breeding garlic specifically to improve peelability, and she told me something I hadn’t previously known: “There’s hardly anyone breeding garlic at all.” Garlic, it turns out, isn’t like other crops, where you plant seeds, grow a plant, harvest it, and then, next year, plant a new seed. Garlic seeds, from fertilized flowers on garlic plants, look a little like onion seeds, but hardly anyone generates and plants them—because why would you? All you need to grow a new garlic plant is just one garlic clove off a garlic bulb. (When you’ve left your garlic sitting around so long a clove sprouts a green shoot, you’ve begun that process.)

No garlic lasts that long around me.

The issue, Kamenetsky explained, is that unlike other crops, garlic mostly can’t flower and be fertilized. “In garlic, this was damaged in ancient times,” she said. “For 5,000 years, people selected for bigger cloves, and they continually selected against flowering.”

Okay, now, see, that's interesting. It's kind of like with bananas. Doesn't have anything to do with a peel, though. Oh yeah, I made that pun.

Within a year of opening, Serafini found a California farm that sold pre-peeled garlic, which is where the Stinking Rose now sources all its A. sativum. “What they do is they put it in screens that heat the skin to a certain temperature, dry it, and then they put it through a wind treatment, like a wind tunnel almost.” The wind blows the skins off. Serafini now swears by pre-peeled garlic: “It’s really the best way! It’s more consistent.”

See? Stop being a snob.

But I hadn’t solved this mystery! My grocery store wasn’t getting some special easy-peel garlic. No one was really breeding easy-peel garlic. So what had happened? It was time to go to the source: Harris Teeter, my grocery chain.

Ah, the problem begins to resolve itself.

The article goes on to describe how garlic shipped from further away has more time to dry and thus might be easier to peel.

I guess I’m grateful that the impossibly convoluted complexity of the intercontinental produce supply chain—which makes modern life more convenient in the short term but is destroying the planet for the long term—is the likely cause of my garlic’s new peelability.

Yeah, at this point, I'll take convenience. We're doomed anyway, so we might as well enjoy the ride. One of the few actual benefits of living in late-stage capitalism is global trade. And the planet's not getting destroyed; only the biosphere.

The result is a system in which buying local, freshly harvested produce can result, bizarrely, in a worse product.

There's nothing at all bizarre about it. If buying local actually gave us better products, we'd never have switched to a global trade model. The only reasons to source local is to support local farms and give yourself something to brag about on social media. The former of which I normally support, except when I know the farmers are voting for the wrong politicians.

But whatever. So, in general, like I said way back at the beginning, the answer is "no, garlic isn't getting easier to peel." Except in some individual circumstances. I'm going to keep wrestling with those little buggers anyway, because the taste is worth it.

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