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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Show Recent Comments

December 14, 2022 at 12:01am

#1041775

Not Buying It

I know I've linked stuff like this in here before, but it's been a while, and not this particular article.

Can money buy happiness? Depends on how you spend it.
Contrary to popular research, people with more money are happier, but it’s their spending habits, not their account balances, that move the dial.

Personally, I don't care if money can buy happiness or not. Happiness isn't, or shouldn't be, the goal. Staying out of misery is a worthwhile goal, and money can absolutely do that.

Happiness is a loving family, a good meal, and an annual salary of $75,000. At least, that’s been the popular wisdom since Daniel Kahneman and Angus Deaton published their 2010 study looking at money’s relationship to well-being. The two psychologists reportedly found that people’s happiness increases until their annual income reaches $75,000, at which point it plateaus.

Given that this was 2010, the threshold is probably up to $100K now. If the study had any merit, which is highly questionable. And also considering that one shouldn't conflate income and wealth.

Like the Milgram experiments, the Stanford prison experiments, and the marshmallow test before it, Kahneman and Deaton’s study is one of few to explode into the mainstream. It has been cited in books, on TV shows, and across social media. CEOs set employee wages to match the findings. And smirking aunts everywhere dragged out the figure to prove that “See, money doesn’t buy you happiness.”

And like all those quoted studies, it has issues. Also, "CEOs set employee wages to match the findings," but, remarkably, the link there only talks about one single CEO who did that. To his credit, he also lowered his own salary to that level. But before you start singing his praises, you may want to look into this a bit further.

But like those other experiments, the popular perception of the research is wrong. Can money buy happiness? No, your aunt is right about that one. But it can facilitate happiness if you spend it thoughtfully.

See, part of the problem here is the way it's phrased: "Money can't buy happiness." That's because once you spend it, you no longer have it. Money, in fact, is happiness.

Just to give one example: Say you're working at a shit job because you need to live and eat and such. Well, what if you got a windfall? Sure, you can keep working. But now you're working knowing that, at any moment, you can up and walk away. That may not fit some peoples' definition of happiness, but it's a hell of a lot less stress.

The article goes on to describe someone who did a different study about the income/happiness crap:

His data showed no obvious plateau at which point money stopped mattering. Happiness rose alongside income all the way up.

I still have issues with how "happiness" is measured. As the article notes, there's at least two kinds: momentary fleeting pleasure, and general life satisfaction. It's all very subjective.

“When you have money, you have options, and that can manifest in different ways,” Killingsworth told CNBC. “Do you buy organic raspberries at the grocery store? Can you quit the job you don’t enjoy, or do you hang on because you can’t afford to be unemployed? … Do you end a relationship with someone that you’re financially entangled with?”

Options. I can see how having the ability to make different choices can contribute to satisfaction. And money is one way to be able to do that.

Money is a medium of exchange after all. It merely stands in for the things, services, and experiences we purchase with it. So when discussing money and happiness, the question isn’t only how much you have. It’s also how you use it.

This tracks. It's obvious that most people wouldn't want to deliberately spend money on things that make them unhappy. What may not be so obvious is that, often, the things you think will make you happy, won't. Like that new iPhone, or an art thing that you just have to have. Those are both destined for the dumpster at some point in the future, when something newer and shinier grabs your interest.

If you spend your money on the things, experiences, and necessities that move your happiness dial, then your money will make your life happier. Not perfect. Not blissful. But happier. If you don’t, then your spending habits can actively work against your well-being and life satisfaction.

Let's be clear, though: spending on necessities shouldn't be a part of that equation. I think the whole thrust of the studies quoted here is: how much does extra income affect happiness once necessities are taken care of?

Another way money facilitates happiness is by bankrolling experiences. People tend to think of money as a way to purchase things, but the joy things provide has a fast half-life. You think that couch is exactly what you need to finally have the living room of your dreams. A few months later, it’s just another thing to sit on.

Like I said.

One reason for this, Norton points out, is that we tend to buy things for ourselves, but we share experiences with others.

I started to snark on this bit, because it seems to exclude introverts. But then I realized: even if I'm not sharing my experiences—road trips, movies, beer festivals, whatever—with someone else in the moment, I do come in here and talk about them (most of the time). That's sharing experiences with others, too.

“Even casual interactions with other people make us happier than sitting by ourselves in a room. So experiences are more interesting and all those things, but they also actually kind of serve to commit us to spending time with other people,” Norton said.

And that's true even for me. Bartenders are cheaper than therapy.

Finally, the more people use their money to give to others, the happier they tend to be. In their research, Dunn and Norton gave participants money to spend in a day. They instructed some participants to spend it on themselves and others to spend it on other people. They found that the charitable groups had a much happier day.

Oh, sure, if you're using someone else's money.

Look, I'm not knocking the act of giving. I've noticed it does provide an endorphin rush, up to a point, and that point is when it starts affecting your own ability to pay for stuff. If you're playing with free money, of course giving it away is going to feel good for most people.

So anyway, I wouldn't put too much stock in this article or the studies behind it. I still firmly believe that "money can't buy happiness" is propaganda to make people feel better about being economically disadvantaged: "I may not have a luxury apartment or a swimming pool or a couple million in the bank, but at least I'm happy!"

If I had to choose only one, money or happiness—I'd pick money. Every. Single. Time.

Happiness can't buy beer.

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