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 20, 2022 at 12:09am

#1036713

Marginal Utility

About a year ago, I commented on a Bloomberg article about dining economics. This entry: "Oh Yeah, I Vaguely Remember "Dates""

. I had to go find it because today's Atlantic article rang a couple of chimes in the old belfry. It's not the same thing, so here is what you might call Dining Economics, Part 2.

The Economic Principle That Helps Me Order at Restaurants
If you’re just eating one dish, you’re missing out.

I order at restaurants using just one criterion: Does it look like I'd enjoy it? The only "economic principle" involved in dining out should be "Can I afford this?"

In the 19th century, when European thinkers began developing the economic principle of diminishing marginal utility, they probably weren’t dwelling on its implications for the best strategy for ordering food at a restaurant.

Probably because they were doing their thinking at taverns, not restaurants.

The basic concept that these early economists were getting at is that as you consume more and more of a thing, each successive unit of that thing tends to bring you less satisfaction—or, to use the economic term, utility—than the previous one.

This may be a valid economic principle (I'll grant that it is), but for me, it simply doesn't apply to dining. My last bite of delicious steak is just as satisfying as the first. The crust of the pizza makes me just as happy as the point. And my third beer may, in fact, be more satisfying than my first.

Recently, Adam Mastroianni, a postdoctoral research scholar at Columbia Business School, invoked this idea in his newsletter, Experimental History, to explain why a flight of beer can be more satisfying than a larger glass of a single brew. “The first sip is always the best sip,” he wrote, “and a flight allows you to have several first sips instead of just one.”

This may be true for others. For me, it's not about diminishing marginal utility. It's about trying as many beers as possible without getting too drunk to enjoy them. Which I suppose plays right into the thesis of today's article, except that I don't have the same desire for variety in food.

The same principle, I’d argue, applies to first bites: If the first half of a dish tends to be more satisfying than the second half, why not have the first half of two dishes instead of one whole dish? In other words, when you go to a restaurant, just share every dish with whomever you’re with. That way, you get more first bites.

But what if they've ordered something you know you won't like?

I've said this before, but just to reiterate: With a few exceptions, such as some appetizers, or pizza, I don't share food. Long ago, I was on a second or third date with someone. On previous dates, she'd pulled the old "I'll just have a salad" thing that some women think all men appreciate, and then proceeded to steal more than half of my fries. So I ordered more fries. Unsurprisingly, this was our last date.

While I'm on the subject of sharing appetizers, can someone tell me why they always come in prime-number servings? Like, five. Or seven. Or whatever number is NOT equal to or easily divisible by the number of diners. It's freaking annoying. Or it would be if I didn't almost always dine alone these days, thus rendering whatever psych trick they're trying to pull invalid.

Diversification can free you from indecision when you’re torn between menu items that sound equally awesome.

For shit's sake, just commit.

For instance, it is the answer to the classic conundrum of brunch: sweet or savory?

What? The classic conundrum of brunch is: beer, mimosa, or bloody mary?

Even I, a prolific meal-splitter, acknowledge that this approach has downsides and limitations. It can be difficult when people have different dietary restrictions or different budgets, and it doesn’t make sense if there’s a dish you know you don’t want to share.

I'm especially not sharing my hamburger. That's gross. Or you have to slice it, which is blasphemy for a burger.

Restaurants do show flickers of awareness that many people don’t want to be locked into eating all of a single dish: They serve buffets, which are basically just meals shared by every customer, and they commonly offer to serve dessert with multiple spoons.

Confession: With the exception of some Indian restaurants, and even then only sometimes, I do not like buffets and would rather opt for menu ordering. No, this isn't a pandemic thing; I was like this in the Before Time.

But a world in which meal-sharing is the default would represent a shift not just in logistics but in values. Whereas a one-dish-per-person paradigm prizes individual choice—and perhaps even endorses a notion of private property—sharing a meal elevates compromise and negotiation.

Big fan of compromise and negotiation, and happy to practice it with something other than my pastrami reuben.

In conclusion, no, economic principles (which may or may not be sound in the first place) don't apply to restaurant dining, except possibly the law of supply and demand.

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