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.

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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.

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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.

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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.

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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

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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

February 14, 2023 at 12:02am

#1044803

Tip, See?

Today, from New York (not to be confused with The New Yorker), an article about the quicksand bog that tipping has become. Some of y'all are going to go out for dinner tonight because of social pressure from some made-up holiday, so it's just as well that this came up at random today. Fortunately, I'm immune. I'll just be staying home and cooking my own dinner, so the only tip I have to give is "don't get sucked in by the commercialism of Valentine's Day."

The New Rules of Tipping

I've tackled this tipping topic before, most recently just last month: "Tipping Point"

In that entry, I pointed out the following:

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.

My opinion hasn't changed in a month and a half.

It is now almost impossible to make any sort of purchase without being confronted with a Square screen asking for 15, 20, or 25 percent.

The previous article said 10, 15, or 20. This may be a New York City thing (today's article is definitely written from a Big Apple perspective), or it may be that it's already creeping up. I've also mentioned before that I've seen much higher "suggestions" in taxi cabs in Las Vegas, so I'm leaning toward the former.

And not just for a coffee: Buying a water bottle at the deli or crackers at a specialty grocery store now sometimes also prompts the option.

Oh, hell no.

This might irritate or confuse you, but the reality is there are new social expectations around what deserves a tip.

I want to know exactly who decides these things. It ain't me.

At restaurants, the previous range of socially acceptable and ethically expected tips was 15 to 20 percent; now, it’s 20 to 25. This goes for whether you’re at an Olive Garden or I Sodi and whether you liked the service or not.

What in the New York hell is I Sodi? (googles) Oh. It's an upscale northern Italian place in Greenwich Village. Look, I don't live there, and even people who do never discover everything about the place. Especially concerning restaurants in the Village, which have a half-life of 2 years.

But that's irrelevant. It occurs to me that the way to end tipping culture, apart from the necessity of pricing things higher and using the excess to pay your wage slaves decent salaries to begin with (said decent salary being significantly higher in NYC), would be not to assert that one doesn't have to tip, but to start a rumor that the only socially acceptable tip is over 100%. If that spreads around, people will freak out and stop going to restaurants, and the entire industry will have to clean up its act or collapse. It's a natural progression of tip inflation, after all. Which is worse than regular inflation, because if you used to tip 15% on a $10 meal, and the meal is now $24 and you're expected to tip 25%, well... (math) your tip that was $1.50 is now $6, a 300% increase on a 140% meal price increase.

Blame this on inflation, COVID, the heightened awareness of the fact that more than half your servers’ salary probably comes from tips. It’s just the rules; don’t complain.

Oh, screw you. Like I said above, the price of the meal is what's affected by inflation. Damn right I'm going to complain. Not to the server, though; I'll still tip and then rant about the entire class-division-enforcing practice right here and to anyone else who'll listen.

At coffee shops, coffee carts, cafés, and bodegas, tip at least 20 percent.

No. Absolutely not. Line in the sand right there. I tip at restaurants because I'm paying for them to take my order, bring the order, bus the table, and generally be of service. Never at coffee shops where you're the one doing 90% of the work, with the other 10% being just like working at McDonald's.

Even though their pay isn’t as tip dependent as waiters’, the average salary for a barista in New York is just above minimum wage.

Sounds like it's time for them to organize.

For food delivery, tip a minimum of $5, or 20 percent, whichever is greater, and even more in bad weather.

On this, I actually agree. Again, they're doing you a service. Would I prefer a system that pays them better where I'm not expected to tip? Sure.

When picking up takeout at a restaurant, it’s easy to understand why you might not tip anything — but you must tip at least 10 percent.

No.

At a bar, the conventional wisdom stands: Tip at least $1 per drink if you’re just getting a beer and 20 percent for a cocktail.

Always tip bartenders.

If you’re at a food counter — a cheese shop, a deli counter, or a fast-casual lunch spot where employees are telling you about the items, slicing, or mixing you a grain bowl — you must tip something if prompted.

I kind of agree with this bit. They're doing you a service, and tips are meant to be for service above and beyond the job description. (I know I've noted this before, but do not be taken in by the myth that TIPS is an acronym.)

Tip Uber drivers the same way you’d tip a cabdriver — at least 20 percent.

I tried tipping a cab driver only 20% in NYC once and got the world-famous New York City attitude for it. Happened in Vegas, too, only worse. But I do tip Uber drivers, for the same reason I tip delivery drivers.

For everything else — hairstylists, waxers, movers — tip 20 percent minimum. And, always, more if you can.

Again, this doesn't seem unreasonable, except that the whole system is unreasonable. It needs to stop, but we're not going to stop it by not tipping; until someone does something to break the cycle once and for all (an apocalypse might do that), I can see the need to tip underpaid service workers.

Just not at coffee shops. Come on.

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