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

September 19, 2022 at 12:02am

#1037860

Putting the Auto in Automobile

Ooooh, scary robot vehicles. Ooga booga.

The big new idea for making self-driving cars that can go anywhere
The mainstream approach to driverless cars is slow and difficult. These startups think going all-in on AI will get there faster.

While the article is interesting and details a different approach to implementing AVs, that's not what I want to talk about.

Instead, I want to talk about how AVs can't get implemented fast enough.

"But Waltz, what about all the stories about autonomous vehicles killing people?" Yes, they happened. Sometimes airplanes kill people, too, and that's considered the safest mode of transportation. Every AV death, and I suspect most injuries caused by them, gets reported. Loudly. Meanwhile, if they did that with every death attributable to human-controlled vehicles? You'd be inundated with horror stories. About a hundred a day, in fact—just for the ones here in the US. One every 10 to 15 minutes, on average. That's just fatalities, not survivable injuries.

Of course, there are substantially fewer AVs than... humVs? Nah, that's already taken. HVs. So you can't really compare them like that. But the point is that, right now, about 40,000 people a year die in HV incidents in the US. While AV technology isn't mature yet, if it can reduce that number by a significant amount, it's a net benefit. What's "significant?" As far as I'm concerned, one standard deviation would do it.

It will not reduce it to zero. That shouldn't be the expectation, or the goal. We've already decided, as a society, that 40K a year is an acceptable death rate for the privilege of driving around. But we also implement things in an effort to reduce that number.

I know I've said stuff like this before, but it's been a while.

There are, of course, significant hurdles to overcome, even apart from the technology needed.

First, public acceptance. As I noted up there, robots are scary. "Common sense" tells you that it's better to have a person in control. Common sense is, as usual, wrong. But that's a big barrier to jump over, considering that many people are scared shitless of airplanes, which are, as I said, about as safe as they come.

Second, and this is even more pernicious, there are a lot of people making a lot of money from human-driven vehicles. I don't just mean Uber drivers, long-haul truckers, cabs, etc.; those are small-time, though that many people losing jobs at once might finally trigger a long-needed revolution. I mean governments issuing speeding tickets, using red light revenue cameras, draining the coffers of drunk drivers (of course drunk driving is bad, but as it stands there are institutions using it as a revenue stream, which means they need it to continue so they can keep getting money), and so on. And AVs don't speed, run red lights, or drive drunk.

Such institutions have a vested interest in keeping that revenue stream coming, so if they haven't already, they're going to campaign against AVs. This won't be limited to lobbying; at this point, I figure any scare article about AVs is being funded by the Fraternal Order of Police or some mayor's office or whatever. No more using speed traps in small towns as a revenue source. No more harassing drivers based on race (unless you count AI as a race). What are cops going to do if they can't pull speeders over? Solve actual crimes? Ha! No money in figuring out who stole your bicycle.

Do things need to be improved before it's implemented? Absolutely. Are there issues to be settled, such as liability, privacy, and how to handle the unexpected? Of course. But the benefits make the work worth it. Well, for other people doing the work, anyway. Me, I'm just hoping the day soon comes when I can take one home from the bar. It's not drunk driving if you're not driving; it's the same as taking an Uber, only without the judgmental glare from the driver's seat.

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