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.

January 21, 2023 at 12:01am

#1043460

Thinking Inside the Box

Science isn't always about probing the origins of the Universe, or figuring out quantum entanglement, or curing cancer. No, sometimes it delves into the most important questions.

You Don’t Know How Bad the Pizza Box Is
The delivery icon hasn’t changed in 60 years, and it’s making your food worse.

I'm not sure that the subhead up there is exactly correct. Yes, as we'll see in this article, the pizza box makes that most perfect of foods somewhat less tasty, but when you consider the extant alternatives, it's really the best we've got.

Where the science comes in is figuring out how to make the best better.

Happiness, people will have you think, does not come from possessing things. It comes from love. Self-acceptance. Career satisfaction. Whatever. But here’s what everyone has failed to consider: the Ooni Koda 12-inch gas-powered outdoor pizza oven.

That's a strong argument, and one I tend to accept, although I don't have one of those.

Since I purchased mine a year ago, my at-home pizza game has hit levels that are inching toward pizzaiolo perfection. Like Da Vinci in front of a blank canvas, I now churn out perfectly burnished pies entirely from scratch—dough, sauce, caramelized onions, and all.

Now I'm hungry. Though that sounds like a lot of work, it's probably one of those few things that are actually worth the effort.

But enlightenment is not without its consequences. The pies from my usual takeout spot just don’t seem to taste the same anymore.

Okay, I'll address the elephant in the room if no one else will: Elephant, why would this guy even bother ordering takeout pizza when he has an Ooni Koda?

They’re still fine in that takeout-pizza way, but a certain je ne sais quoi is gone: For the first time, after opening up a pizza box and bringing a slice to my mouth, I am hyperaware of a limp sogginess to each bite, a rubbery grossness to the cheese.

You don't have to have three and a half years of Duolingo French lessons under your belt to know what "je ne sais quoi" means: "I don't know what." In the rest of the article this author asserts that he does, in fact, know what.

Pizza delivery, it turns out, is based on a fundamental lie. The most iconic delivery food of all time is bad at surviving delivery, and the pizza box is to blame.

One of my favorite breweries is right here in my hometown. During the lockdown in 2020, I supported them by ordering beer and food for delivery about once a week. Canned, or bottled, beer, isn't as good as draft, but it's not bad. Their burgers survived the 2-mile delivery trip quite well. Their frites, however, arrived soggy and mushy; they're much better if you get them at the restaurant. They put a bunch of frites in a little metal basket, which gets dipped into the fryer oil and delivered, basket and all, to your table. Naturally, the basket doesn't come with the delivered version, which is instead handed to you in a recycled-cardboard container.

While this particular brewpub doesn't do pizza, the frites thing is a close equivalent to what this author is talking about.

A pizza box has one job—keeping a pie warm and crispy during its trip from the shop to your house—and it can’t really do it.

Warm, sure, to an extent. That corrugated cardboard is pretty good insulation. As he describes later, though, that same box concentrates moisture inside, turning the pizza limp.

The fancier the pizza, the worse the results: A slab of overbaked Domino’s will probably be at least semi-close to whatever its version of perfect is by the time it reaches your door, but a pizza with fresh mozzarella cooked at upwards of 900 degrees? Forget it. Sliding a $40 pie into a pizza box is the packaging equivalent of parking a Lamborghini in a wooden shed before a hurricane.

I don't think I've ever ordered a $40 pizza. Sometimes, by the time delivery fees and driver tips are included, I've come close... but never quite $40.

I know for a fact I've never had a Lamborghini, or a wooden shed.

And yet, the pizza box hasn’t changed much, if at all, since it was invented in 1966.

This is probably due to economics. But this is where the science comes in. Or, perhaps, engineering, which is really just applied science: come up with a pizza delivery system that keeps the pie warm but doesn't ruin it, and doesn't cost much. As noted above, Domino's, probably the largest chain, has no incentive to do this; their shit is shit whether it's "fresh" or out of a delivery box. So it's going to be up to actual scientists and/or engineers. Unfortunately, while this article is very descriptive, it doesn't propose actual solutions.

To be fair, neither can I. I just want my pizza.

Unlike a Tupperware of takeout chicken soup or palak paneer, which can be microwaved back to life after its journey to your home, the texture of a pizza starts to irreparably worsen after even a few minutes of cardboard confinement.

If you reheat it right, though, leftover pizza can be delicious. I know I've linked to some scientific experiments along those lines in here before. Ah, here it is, from October of 2021: "What Do You Mean, "Leftover Pizza?""

That discussion doesn't address the problems with the pizza box, though.

The basic issue is this: A fresh pizza spews steam as it cools down. A box traps that moisture, suspending the pie in its own personal sauna. After just five minutes, Wiener said, the pie’s edges become flaccid and chewy. Sauce seeps into the crust, making it soggy.

Worse, the poor benighted souls who have never ordered pizza from an actual New York City pizzeria and eaten it right there on the spot think that this is what pizza is supposed to taste like.

By 1949, when The Atlantic sought to introduce America to the pizza, the package was already something to lament: “You can take home a pizza in a paper box and reheat it, but you should live near enough to serve it within twenty minutes or so. People do reheat pizza which has become cold, but it isn’t very good; the cheese may be stringy, and the crust rocklike at the edges, soggy on the bottom.”

What I didn't note is that today's article is also in The Atlantic.

Corrugation produces a layer of wavy cardboard between a top and bottom sheet, sort of like a birthday cake. The design creates thick, airy walls that both protect the precious cargo within a pizza box and insulate the pie’s heat while also allowing some steam to escape.

I should note that I have gotten takeout pizza (if not delivery) that was packaged in a single-ply, though thick, cardboard box. It's not any better at keeping the pizza at peak.

We’ve gotten a couple of pizza-delivery innovations in the past few decades: the insulated heat bag—that ubiquitous velcroed duffel used to keep pies warm on their journey—those mini-plastic-table things, and … well, that is mostly it.

I've actually had people ask what the table is for. That's okay; it's not necessarily blindingly obvious. It's to keep the top of the box from contacting the toppings, and potentially pulling them off. Then you have a pizza crust, and a cardboard box top with the toppings on it. Which, to be fair, wouldn't taste much different from Domino's.

“Every single pizza that I put in a box I know is going to be, let’s say, at least 10 percent not as good as it could have been,” Alex Plattner, the owner of Cincinnati’s Saint Francis Apizza, told me. Others dream of better days. “After smoking a lot of weed, I have come up with a lot of ideas for a better box,” said Bellucci, the New York City pizza maker.

Weed is legal for recreational use in New York City now, so there should be a slew of innovative ideas coming out of that metropolis any day now. Ideas, but not necessarily their execution. Too much work for a stoned person.

And I just have to say how hilarious Saint Francis Apizza is.

Last year, the German brand PIZZycle debuted the Tupperware of pizza containers, a reusable vessel studded with ventilation holes on its sides.

I take back the bit about weed. If it's going to lead to people naming their brand PIZZycle, maybe we should stick to booze. No, there's no evidence that weed was involved in that decision, but there's a strong link between pizza and getting stoned, so I assume the connection in the absence of evidence to the contrary.

So we know it’s not a question of ingenuity: We can construct better pizza boxes, and we already have. The real issue is cost.

Like I said.

Domino’s alone accounts for nearly 40 percent of delivery-pizza sales in the U.S.—on par with all regional chains and mom-and-pops combined. Perhaps these big companies are stifling real pizza-box innovation.

I shouldn't be surprised. This is the same "culture" that insists on soft white bread, pasteurized process cheese "food," and rice-adjunct lagers. We, as a society, have crap taste. I don't personally like chicken wings, but when spicy chicken wings became popular, I at least held out some hope that we'd get over our phobia about any spice hotter than mayonnaise, but that hasn't happened.

Again, though, if you have your own backyard gas-powered 900 degree pizza oven, why are you even bothering with delivered pizza? I mean, I'm all about lazy, but pizza transcends even that.

Now, if you'll excuse me, I have a frozen pizza to bake.

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