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.

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.

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

November 4, 2022 at 1:54am

#1040225

Monkey!

I ever tell you why I don't like gambling in Atlantic City?

Yeah, yeah, I know, I do it anyway. But I get to bitch about it if I want.

When I play blackjack, I generally prefer the second chair. Picture a casino table, right? Dealer on one side, six player positions on the other. Chairs are numbered from the dealer's left, not the players' left. Sometimes it's called second base, which is weird when you think about it because there's almost always six chairs at a table. Anyway, I'm not that picky about it, but if I have a choice I'll take second so I don't have to deal with the idiot decisions of too many people to my right. More about that later. The dealer proceeds from their right to left, so if all the seats are filled, the deal goes: one card to 1, 2,... 6, dealer up card, second card to 1, 2,... 6, dealer down card. Then the players make their hit/stand/double/whatever decisions also in order 1, 2,... 6.

Like any pastime, blackjack has its own insider lingo. Probably everyone knows the basics: hit, stand, double down, split. One particularly annoying piece of lingo that I think is of more recent origin is "monkey" for any card worth 10 points (ten, J, Q, K). This apparently originated in Asia, where presumably the word was in a different language and maybe it has an association with the number 10 in one of those languages. Or maybe a monkey symbolizes luck. I don't know.

But Americans picked that shit up and overused it. So you're sitting there at a table, getting into the groove, when some lily-white trust fund brat chips up. He gets dealt an ace. Slap the table, "MONKEY!" Then, next hand, he gets a four-seven. Double down and slap the table. "MONKEY!" Next hand, dealer's showing a six, flips another six. Slap the table. "MONKEY!"

On that last one, it's almost worth watching the dealer draw a nine for a total of 21 just to listen to the kid moan about it.

Point is, for fuck's sake, mix it up a little, okay?

To be fair, this isn't an AC problem; it happens everywhere I've played blackjack.

Then you get the ten-splitters. Which I do see more of in AC than in Vegas.

There's nothing inherently wrong with splitting tens, but the odds aren't in your favor if you do that. Let me see if I can explain without too much math. So you get dealt two monkeys (ugh). That's a 20. In something like 9 out of 10 hands, that's at the very least a push, if not a win. There are no sure hands in blackjack (even the namesake hand could be a push), but a 20 is pretty good odds, no matter what the dealer's showing (if they have 21, they show before you get a chance to split, and you lose anyway unless you have 21 also). And yet you get greedy people who split tens. They're then basically playing two hands. You can split again. In AC, you can have as many as 4 hands from splits. it's rare. Now, on each 10, you've got a 4/13 chance (roughly) of getting another 10, and a 1/13 chance of each other card. It could even be an ace. But that ace doesn't give you blackjack; that's only a thing on your first two cards. It's still 21, guaranteed to at least push (tie with the dealer). But again... low odds of pulling an ace.

Some players think it messes up the odds for the other players for someone to split tens. I disagree. Decks are thoroughly shuffled, and that next card could be anything (I'm talking about multiple-deck blackjack here). But it's maddening as hell if you've got, say, an 11, and you want to double down, and you watch the ten-splitter to your right keep getting monkey after monkey when the card you want is a monkey, and when they're finally done you end up with a five (that gives you 16, which sucks ass, especially after doubling).

I mean, sure, it can also go the other way, but it's still annoying, and that's why I like to sit at 2; it minimizes the risk of people to your right making questionable choices.

Also, some people like to play two spots. That's fine; it's allowed, with dealer permission, when things aren't too busy. I've done it myself on occasion. So yesterday, I'm sitting there at second, and there's a couple at 4 and 5, leaving open only 1, 3, and 6. Some dude comes up and sits next to me at 1. "Hey man, can you scoot over? I wanna play two hands." Now, look, changing seats at a blackjack table is a Big Damn Deal (no pun intended for once). The dealer probably doesn't care; they just slap cards down at whatever spots have chips in 'em, but the pit bosses are all keeping track of your betting patterns and whatnot. This isn't as intrusive as it sounds; come on, you're in a casino and there are cameras everywhere already. They do this to keep track of your points for comps and tier credits or whatever the casino uses to determine if you're spending enough money to earn Free Parking (that is a pun; Monopoly was based on Atlantic City).

Being a relatively nice guy, I'm fine with switching seats so the one dude can play two spots. So I go through the hassle of moving my beer, my chips, my cigar, and my secret card counter (just kidding about the last bit) one seat to my left. Game pauses while the pit bosses mess around with the touchscreen to sort everything out. Guy finally settles in and throws a chip at 1 and 2, and also plays the sucker bet on both.

He loses.

He gets up and walks off.

Look, blackjack is a grind, not an instant jackpot. More often than not, you're going to lose the first hand. Sometimes, over time, you can amass some winnings, but your chip stack is going to go up and down like the Apple stock ticker in the meantime. That is, if you're not me and you're not playing in Atlantic City. I have better luck playing the game in Vegas, for whatever reason (the rules aren't significantly different, so that's probably just chance, or possibly confirmation bias). On the other hand, for some reason, I do better at slots in AC. Go figure. Slot machines are heavily house-weighted (blackjack is only slightly house-weighted), so I don't play them often, but for some reason, yesterday, I won back most of my blackjack losses by hitting a (very minor) jackpot on a slot.

That's when I quit gambling for the week. I'm leaving today anyway, so no big deal.

With all that, why do I keep going back here? Well, I had a vague memory of talking about AC in an earlier blog entry, so, being done with gambling for this trip, I had time to look it up, from way back in 2018: "Atlantic City"

Excerpt:

"But they still have bars, and blackjack, and usually you can avoid the scam artists wandering in off the streets for long enough to let casino security kick them out. And, unlike Vegas or Reno, if you look out the window you can sometimes see the ocean, and forget for a moment that those waves will one day drown the city in a flood of sharks and used needles.

Sometimes the only purpose of a place is to remind you of how good you actually have it."

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