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.

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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.

January 5, 2023 at 1:34am

#1042667

Tipping Point

Reminder: entries here will be at irregular times for the next week or so, while I'm away from home.

Appropriately enough, today's article, from CNN, is about tipping.

‘Out of control’: No one knows how much to tip

Coincidentally, the last time a tipping article popped up out of my queue was also the day I was scheduled to leave on a trip: "Hot Tip"

A new checkout trend is sweeping across America, making for an increasingly awkward experience: digital tip jars.

Article is from last month. Last. Month. "New trend?" I don't get out much, and I noticed this "new" trend several years ago.

You order a coffee, an ice cream, a salad or a slice of pizza and pay with your credit card or phone.

How dare they. How DARE they not mention beer.

Then, an employee standing behind the counter spins around a touch screen and slides it in front of you. The screen has a few suggested tip amounts – usually 10%, 15% or 20%. There’s also often an option to leave a custom tip or no tip at all.

Yes, that sucks. It can feel like begging. But I want you to keep in mind that, as noted, you do have the option for no tip, however embarrassing you might think that is. That's a you thing.

The worker is directly across from you. Other customers are standing behind, waiting impatiently and looking over your shoulder to see how much you tip. And you must make a decision in seconds. Oh lord, the stress.

Someone sounds paranoid.

Do you tip at Burger King? No? How about at a movie theater concession stand? No? The cash register at the 7-11? No? Then why be expected to tip at a self-service coffee shop?

Sure, if someone goes above and beyond, maybe. In situations where you're seated and a server takes care of you and someone cleans the table before you arrive and after you leave, absolutely.

Now, before I get shit from foreigners here, yes, I'm fully aware that American tipping culture is fucked up in the first place. Whole etiquette articles are written about it, and no one seems to agree on anything. Like the one I linked in the entry above. It's a maze of unwritten rules and expectations; it's vaguely racist, absolutely classist, and seemingly designed as one enormous shibboleth to weed out foreign tourists. It reminds me of the convoluted unwritten but absolutely essential bribery rules in other countries. I'd like to see it go away almost entirely, with prices raised across the board so servers are making a decent wage without the need for tips.

Until that happens, if civilization doesn't collapse first, we're going to have to deal with this sort of thing. Once a complex set of rules exists, people who have mastered it (or in this case, think they have) don't want to see it go away. See also: US income tax filing, English grammar and spelling.

Although consumers are accustomed to tipping waiters, bartenders and other service workers, tipping a barista or cashier may be a new phenomenon for many shoppers.

Like I said, not new. Oh, by the way, always tip bartenders. What's the difference between a bartender and a barista? Well, for starters, the barista usually doesn't have to put up with your drunken opinionated rambling. That alone is worth a tip.

“I don’t know how much you’re supposed to tip and I study this,” said Michael Lynn, a professor of consumer behavior and marketing at Cornell University and one of the leading researchers on US tipping habits.

At least, unlike the "expert" interviewed in the previous tipping article I shared, this professor doesn't seem to have fallen for the fauxtymology of tips being an acronym.

Adding to the changing dynamics, customers were encouraged to tip generously during the pandemic to help keep restaurants and stores afloat, raising expectations.

Nonsense. Unless the restaurant is breaking the law, the tip should go straight to the worker, never to the restaurant's bottom line.

Customers are overwhelmed by the number of places where they now have the option to tip and feel pressure about whether to add a gratuity and for how much.

I treat it like I do the begging for change "for the children" at any checkout: ignore it.

“If someone can afford Starbucks every day, they can afford to tip on at least a few of those trips,” added the employee, who spoke under the condition of anonymity.

I need to find or create a name for this fallacy. I see it everywhere, in different contexts. Like "If you can afford a Porsche, you can afford the insurance on it." This is absolutely illogical and it's utter nonsense. It also leads to an infinite sum. "If you can afford A, you can afford B. If you can afford A+B, you can afford C. If you can afford A+B+C, you can afford D."

Tipping spread after the Civil War as an exploitative measure to keep down wages of newly-freed slaves in service occupations.

Hence my "vaguely racist" comment above.

Critics of tipping argued that it created an imbalance between customers and workers, and several states passed laws in the early 1900s to ban the practice.

Hence my "absolutely classist" comment above.

Just how much to tip is entirely subjective and varies across industries, and the link between the quality of service and the tip amount is surprisingly weak, Lynn from Cornell said.

He theorized that a 15% to 20% tip at restaurants became standard because of a cycle of competition among customers. Many people tip to gain social approval or with the expectation of better service.

It's not "entirely" subjective. Most everyone in the US knows that the standard tip for restaurant servers is 20%. This, however, is a new development; when I was a kid, the standard was 15%. Now you get shamed if you tip only 15%.

As for the "expectation of better service" thing, think about it. You tip at the end of the meal, when the service is complete. If you're a regular, sure. If they remember you next time. But if you're in another town, that's not an incentive. You can walk out without tipping, and there will be no repercussions to anything but your conscience. No, you do it so the next schlub doesn't get a surly waitress.

An MIT study released in 2019 found that customers are less likely to tip when workers have autonomy over whether and when to work. Nearly 60% of Uber customers never tip, while only about 1% always tip, a 2019 University of Chicago study found.

Well, I'm in that 1%. Unless the Uber driver crashes or tries to convert me to their religion. Neither of these things have happened yet.

Etiquette experts recommend that customers approach the touch screen option the same way they would a tip jar. If they would leave change or a small cash tip in the jar, do so when prompted on the screen.

So, no. Got it.

Okay, so maybe this wasn't the best article to pop up while I'm trying to pack for my trip. I tend to get ranty about it.

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