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

October 27, 2022 at 12:01am

#1039805

Battle of the Budget

I think it's been a while since I talked about this sort of thing.

An economist studied popular finance tips. Some might be leading you astray

Oh, an economist. Well, there's an exact science for you.

Managing your money is obviously an important part of being a responsible adult.

Huh. I wonder what the other parts are?

It turns out that there's a large gulf between the advice given by the authors of popular finance books and academic economists.

Why that might be, I leave as an exercise for the reader.

In a new study titled "Popular Personal Financial Advice versus the Professors," the Yale financial economist James Choi rummages through 50 of the most popular books on personal finance to see how their tips square with traditional economic thinking.

There are more than 50 of them?

It's like a cage match: Finance thinkfluencers vs economists dueling over what you should do with your money.

Never. Ever. EVER. Use that word again.

Traditional economic models portray humans as hyper-rational, disciplined creatures, who always make optimal financial choices for themselves.

Ha! Hahahahaha. Ha.

Behavioral economics, which has pretty much taken over the field, emphasizes that people are quirky, often irrational, and prone to errors.

Great! All the best parts of psychology and economics, all rolled up into one tight package!

But, Choi says, the advice of popular finance thinkfluencers, who tend to concentrate on helping us overcome our flaws and foibles, might actually be more effective in some cases.

I'mma smack someone.

When it comes to saving money, many economists offer somewhat counterintuitive — and, dare I say, potentially irresponsible — advice: if you're young and on a solid career track, you might consider spending more and saving less right now.

Sure. You're going to do that in any case. Might as well help boost end-stage capitalism while you're at it. We don't have a future, anyway.

The idea, Choi says, is "you don't want to be starving in one period and overindulged in the next. You want to smooth that over time."

Except if you don't want to work. Then you work your ass off, eat noodles, live in a box, and save every penny you can to hasten the day that you can tell your boss(es) to sod off.

"I tell my MBA students, 'You of all people should feel the least amount of guilt of having credit card debt, because your income is fairly low right now but it will be, predictably, fairly high in the very near future,'" Choi says. Once they start making money, he says, they should probably pay down that debt quickly since credit card companies charge high interest rates.

Again, that's what people do without the wonderful advice of professors.

Where thinkfluencers and old-school economics really depart from each other, Choi says, is "the usefulness of establishing saving consistently as a discipline," Choi says.

Why am I still reading?

In old-school economics, money is money. It's fungible. There is no reason to put labels on it. Absent some financially advantageous reason to do so (like the ability to get subsidies or a lower tax rate), it doesn't make sense to set aside savings for specific purposes, like a new car or a future vacation or a down payment on a house. A dollar is a dollar.

Then why even bother with a fucking budget? A budget is all about assigning categories to lumps of money.

Choi finds that 17 of the 50 books he read through advocate for some sort of mental accounting exercise. And, he says, this advice might actually make sense.

It should be 50 out of 50.

Many Americans live in enormous houses and are stretched thin paying for them. While their house is a valuable asset, and they're technically pretty rich, they're just squeaking by, living paycheck to paycheck. People generally refer to this as "house rich, cash poor."

Somehow, we've bought into the idea that your house is an investment. It is not. I've spent more money upgrading, fixing, and renovating my house than I spent on the goddamned house, including the interest on the various mortgages I've had on it. And then what am I going to do, sell it? Then where do I live? A rental, which is even more of a money suck?

Also, it seems that some people think that the only alternative to an enormous suburban house is a trailer (or they call them "tiny houses" or something these days.) People don't consider the middle ground very often.

Generally speaking, popular financial advisers say that, while stocks are risky in the short run, you should invest mostly in them when you're young because they earn higher returns than bonds over the long run. "The popular belief is that the stock market is kind of guaranteed to go up if you just hold onto it for long enough," Choi says.

Stocks and bonds, however, are investments. But no, there are no guarantees.

But while popular authors may discount this risk over the long term, their advice recognizes that holding stocks is risky in the short term.

I'd even go so far as to say that it's gambling in the short term. Nothing wrong with gambling, in my view; just know that you're doing it.

"For almost all working people, the major economic asset they have is their future wage income," Choi says. In other words, think of your work skills (your "human capital") as part of your financial portfolio. It's like the biggest form of wealth you own, and it's generally safer than stocks or even bonds.

Yeah, no. You still have significant risk if you look at it like that. If you're injured or ill, for example, your earning potential decreases.

The rest of the categories here have to do with tweaking stock and bond investments. Worth a read even if you don't do that. But I'm not going to quote any more from the categories.

So who wins? The thinkfluencers or the economists?

GAAAAH. Where's my tequila?

Who wins? Why, whoever can sell the most books, of course.

"I think of it in terms of diet," Choi says. "The best diet is the one that you can stick to. Economic theory might be saying you need to be eating skinless chicken breasts and steamed vegetables for the rest of your life and nothing else. That's going to be the best for your health. And, really, very few people will actually do that."

I think I've noted before the similarities between budgeting and dieting. Or, as I prefer to think about it, having a money plan and an eating plan (those don't have the same implications of self-denial). Oddly enough, I'm pretty good with the money plan. Eating? Not so much. Now, where did I leave that pizza? Oh, yeah. Right next to the tequila. How convenient.

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