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

June 13, 2023 at 8:09am

#1051013

A Whole Lotto Nonsense

Today's article, from Slate, is almost a year old. Still relevant, though, apart from a few details.

Why the Myth of the Miserable Lottery Winner Just Won’t Die
Actually, most people do just fine with their millions of fresh new dollars.

On Friday night, Mega Millions held a drawing for a whopping $1.28 billion jackpot, the third largest in American history.

That's the no-longer-relevant detail.

Other sources

put it at 1.337 billion, which I find amusing because 1337 is computer nerd code for "leet," meaning elite, meaning awesome.

Also, it turned out that the ticket was redeemed anonymously, apparently by a two-person partnership. Some lotteries let you do that. Should you ever win the lottery, I highly recommend going that route, tempting as it may be to shout your good fortune from the rooftops.

One in eight American adults play the lottery at least once a week, and almost half buy at least one ticket a year.

I'm a gambler, but I don't buy lottery tickets. Because I do play other games of chance, I don't heap scorn upon those who do. Everyone has their thing.

Unfortunately, even if you bought your annual ticket sometime in the last few days, that winner is probably not going to be you—unless you happen to be that lucky Illinoisan.

Obviously, this article was written before the anonymous partnership stepped forward to claim the winnings.

Maybe that’s okay, you tell yourself. Don’t lottery winners end up broke and miserable? It would be great to be rich, but I don’t need $1.1 billion.

No one "needs" $1.1 billion. I wouldn't turn it down, though. However, the risk/return ratio on a lottery is too high for my taste.

Except most lottery winners do not wind up broke, or miserable, or bankrupt.

This is the important part of the article, as evidenced by the headlines.

Lotteries have been characterized as a tax on people who are bad at math. I can't fully disagree with that, though it's not an absolute certainty that if you play the lottery, you're bad at math.

What is probably true is that if you were bad with money before you win the lottery, you'll be bad with it afterward, as well. Money fixes a lot of things, but carelessness isn't one of them.

However, in this particular case, everything about the outcome (as reported in the second link above) signals that the winners made good decisions:

- Forming a partnership, which most likely clearly spelled out who would get how much of the winnings;

- Remaining anonymous;

- "Working with professional legal and financial advisors to support the claim process."

Stories about regretful lottery winners are trotted out whenever jackpots get big. But as much as jealous losing bettors might want to think that winners’ unfathomably good luck is balanced out by bad, most people who strike it rich this way settle into lives of quiet, comfortable anonymity.

You only hear about the bad-luckers, much as you generally only hear about the few planes that crash, as opposed to the millions that take off and land without major incidents.

And yet, the myth of the miserable lottery winner persists. The history of this myth reveals a longstanding national discomfort with gambling, and exposes deep-seated cultural beliefs about the connection between wealth, work, and merit.

And as with plane crashes, this sort of thing does happen, and with far more frequency than plane crashes. But, again, it's hardly a certainty.

From the 18th through the 20th century, newspapers in the United States recounted the misfortunes of lottery winners from across the globe: A baker and his pregnant wife murdered for his winnings by an employee (Paris, France, 1765). A squandered jackpot invested in a failed shipping venture (Newburyport, Massachusetts, 1883). A winner dying of a heart attack immediately upon hearing news of his windfall (Bilbao, Spain, 1934).

Isn't that ironic?

University of Buffalo sociologist H. Roy Kaplan interviewed around 100 early lottery winners in the mid-1970s and found most of them happy, despite these challenges. Nonetheless, a narrative was born.

I can't really blame journalists, though. It's a well-known truism that bad news sells better than good. "Man wins lottery, gets robbed" is inherently more interesting than "Man wins lottery, keeps working, dies happy."

Abraham Shakespeare was killed by an acquaintance three years after winning $30 million in 2006.

I'm just including this quote because that dude had an awesome name, worthy of a $30 million win.

As an aside, yes, I know that the actual payouts are less than the stated payouts. The $30 million (or $1.337 billion or whatever) is the present value of a 20-year annuity; one usually has the option to take it as a lump sum up front, which reduces the payout by the value of the annuity; and taxes (at least in the US) are automatically withheld at delivery. But the stated payout numbers are still relevant, because it's a way to compare the different jackpot sizes directly. Kind of like how a window sticker announces a car's gas mileage at the dealer, but no one expects to actually get those numbers on the road.

Their stories are repeated so often not because they are representative but because they are some of only a few examples of regretful winners. The vast majority of jackpot recipients collect their novelty checks at press conferences and are never heard from again.

That may not be the best phrasing. "Never heard from again" can also imply someone's gone missing or dead.

Research into winners in Germany, Singapore, and Britain found that winning the lottery does, in fact, make people happier, and a 2004 study found that 85.5 percent of winners in Ohio kept working, a sign of how many carried on with their normal, pre-jackpot lives.

Some people actually like working. Not me. But some people. I knew a surveyor who had won a lottery; the payout in his case was "only" about a million bucks, but this was back when a million bucks meant something.

Money, it seems, really can buy happiness.

I joke a lot about how money can't buy happiness, but it can buy beer, and that's good enough for me. But on a more serious note, the way that old cliché is phrased is a problem: it implies that you trade money for happiness (because when you buy something, you trade money for it). This is rarely the case. On the other hand, having money can result in a peace of mind and contentment, which is basically happiness.

Why, despite all the available evidence, does the myth persist? What does it mean that this narrative is believed so widely?

The article goes on to answer those questions, though not in as detailed a way as I would have liked.

But Americans’ enduring love of gambling has long been in conflict with an important element of the nation’s mythology: that the United State is a meritocracy founded on hard work, a place where the smart, the savvy, and the deserving rise to the top, no matter their background. The implication of this ethos is that hard work always yields a just reward. By design, the meritocracy leaves little room for chance.

What they leave out is something that is self-evident to me: that this "mythology" is a myth in both senses of the word: a foundational story, and a fiction. It's abundantly clear that hard work does not automatically lead to financial success; if it did, sharecroppers would be millionaires. Plenty of the "smart, savvy, and deserving" are living paycheck to paycheck, while others make their millions, or more, by cheating and dodging (which, again by American foundational ethics, are no-nos).

Also, even people who come by their fortunes "honestly" (hard work, etc.) can often lose it all: sports idols, rock stars, business owners, etc. And it's not always drugs or profligate spending that brings them down; here in the US, sometimes it just takes one visit to the emergency room to go bankrupt.

I've written in here before about the role of luck in a person's fortune (financial or otherwise); this is no more or less true with regards to lottery winners.

A lottery winner is no more or less deserving than a company CEO or a Hollywood superstar. Sure, the latter two presumably put in the work, but they were also lucky enough to have the talent to do whatever it is that made them money. In truth, the concept of "deserving" is a social construct, a story we tell ourselves.

And in the end, just as good luck can happen to any of us, so can bad (the difference, of course, is subjective). And we all die in the end, so why not enjoy things while you can? If that means, to someone, playing the lottery and dreaming of a better or easier life, let them have the fantasy.

I'd be more focused on how the government is spending their share. You hardly ever see stories about that.

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