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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July 28, 2022 at 12:03am

#1035779

Enough

They say money doesn't buy happiness. I disagree. It buys beer, and beer is happiness.

How Much Money Is 'Enough'? This Simple Thought Experiment Gives You an Exact Number to Aim For
Constantly chasing more and more will make you miserable. The right goal gets you off the treadmill.

This financial article has been lazing around in my queue, not even earning interest, for a long time. It's a couple of years old, but always relevant (at least until our inevitable doom).

Have you ever read those articles where some extremely well-off family details their budget and then bemoans that they're barely getting by?

Yes, and I've also read ones that talk about how a young couple managed to buy (gasp) a house in this economy. Always, without fail, their technique involves having rich parents who help out.

It's ridiculous that anyone could complain about raking in $350,000 a year, and it's clear many of these folks are wildly out of touch with how privileged they are.

Or it would be ridiculous, except that it seems like most people think "rich" means "making lots of money." It does not. It means "having money." It does no good to be making $350,000 a year if you spend all of it (and definitely if you spend more than that). Someone making $50K and saving $10K of that is richer; they're just not going to have nearly as many toys.

It's not just the wealthy who fall into the trap of earning more only to spend more and feel just as dissatisfied.

How do you get off this treadmill?

Most people also think of a treadmill as a means for getting fit and/or controlling one's weight. It was originally designed as a human-powered source of work,

and later repurposed for punishment.

Research shows that up to a certain threshold (studies consistently put it at about $75,000 dollars a year, give or take a bit depending on cost of living) money has a big impact on both day-to-day happiness and life satisfaction.

I've never been entirely convinced by those studies. Not that I think they were poorly designed, but a) that's an average, so some would settle for less while others need more to hit that threshold; and b) again, it's not what you make, but what you save.

But perhaps the best way to get a feeling for your goal number isn't math but a simple thought experiment from writer Brad Stollery:

Before I paste that thought experiment here, I'll add something else. The article points out that accumulating about 25 times your expected annual expenses is the path to financial independence. I'm not going to argue that; it tracks with the common advice to live in retirement off of 4% of your liquid net worth. Fine. I will say, though, that it's important to decide if, when you reach that savings goal (if you ever do), whether enough is actually enough. You find yourself in retirement, especially a theoretical early retirement, and suddenly you have all the time to do all that traveling you never got to do because vacations were discouraged at work—but traveling ain't cheap.

I won't paste the entire thought experiment, just the—pun intended—money quote:

How much money would you have to be paid, right here and now, to retire today and never receive another dollar of income (from any source) for the rest of your life?

The catch this time is that whoever among the five players writes the lowest amount on the check will be paid that sum. The other four players will get nothing.

Now, I'm well aware that some people actually enjoy working for a living. I do not. It gets in the way of my video gaming. If you're one of those people, this article is not for you—but I still think it's important to have money stashed away in the event of disability or extended job loss.

This thought experiment forces you to cut away the natural impulse to aim ever upward (if you do that you'll bid too high and get nothing). That result is however much you ask for is your number, the amount you'd need to live comfortably and pursue your goals if status and lifestyle inflation weren't a factor.

Your expenses and desires can be infinite. If you don't want to chase them miserably forever, you need to put a cap on your financial ambitions yourself.

I dunno; do the famous billionaires (you know who they are) seem stressed or miserable to you?

On that note, there's a cultural bias surrounding the nice round sum of one million dollars (obviously, this has an American slant). "If I had a million dollars" isn't just the name of a catchy song by Barenaked Ladies (who are from Canada but the point remains); it's the number a lot of people have ingrained in their heads as the cutoff between "rich" and "well-off."

But even before this year's higher inflation, a million bucks wasn't what it used to be. Using the 4% rule I talked about earlier, that translates to an income of $40,000 a year—roughly what a $20 an hour job pays.

On the one hand, if you reverse that math, if you're making $20 an hour, then your effective net worth is $1M. On the other hand, $40K annually doesn't do much these days. It's just slightly more than enough to cover the basics in most places in the US, and actually less than you'd need to live in one of the big coastal cities.

There's another aspect that the article doesn't cover, and that's taxes. Since state taxes vary (and don't exist in some states), you'd have to do your own research there. (This does NOT mean looking at YouTube videos until you find one that tickles your confirmation bias.) But the most effective place to keep a million bucks is in the stock market. Sure, it has its ups and downs, but over a long enough time frame, it usually beats inflation. And here's the thing about stocks: realizing long-term capital gains usually results in a lower tax burden than working for a living. I mention this because it's possible to have a lower overall tax burden if you're living off of saved money, which means you might not need as much as you think you do. You'd have to look into that yourself, though.

Lots of people seem to have a problem with this, but remember, the money you put into securities (apart from certain retirement accounts) has already been taxed.

Okay, I'll shut up now. Mostly I think I just saved the article because I thought it was a different way than usual to look at things. And of course I had my own commentary to add.

Also, since I mentioned a certain song, and YouTube:

Haven't you always wanted a monkey?

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