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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QPdoll

about 10 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 13 hours ago
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Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
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WakeUpAndLive️~Happiness

a day ago
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Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

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https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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October 7, 2023 at 9:13am

#1056913

Autonomy

It doesn't matter how safe they are; there are those who are scared shitless of them, and no amount of logic or number-talk will change their minds.

Are self-driving cars already safer than human drivers?
I learned a lot by reading dozens of Waymo and Cruise crash reports.

Because I don't have a vested interest in anything, I'll say up front that the answer to the headline question turns out to be "maybe."

There's not a lot I need to quote from the article; it's there if you want to read it. I doubt it'll convince anyone of anything, though.

Neither will I, but I'm going to talk about the subject anyway. I've done it before, in "Putting the Auto in Automobile"

, and others, and I'll try not to repeat myself too much.

Here's the important quote:

Of course self-driving cars are flawed—all technologies are. The important question is whether self-driving cars are safer than human-driven cars.

And that's where data analysis comes in. The issue, and this is touched on in the article, is that AV incidents are scrutinized to a much finer level of detail than all but the worst HV (human-driven vehicle) incidents, and every little bump and ding is subject to this scrutiny. Imagine if you had to fill out incident reports and have questioned every decision you made leading up to the event every time you hit a squirrel in the road, or dinged someone's bumper in a parking lot.

Point is, accurate comparisons are hard. And while traffic fatalities are, in one sense, common (usually on the order of 40,000 deaths per year in the US), in another sense, they're pretty rare: according to the linked article, an average of one per 100 million miles driven.

And this is just me, but I'm not sure if we can take fender-benders as a proxy. That is, you expect x number of serious injuries for y number of minor incidents with human drivers, and the ratio of x over y is probably fairly constant year to year. Human drivers make mistakes. Robots make different mistakes.

On the other hand, we can't just set the robots loose, see how many people they kill, and then make a decision. That would be slightly unethical. No, we have to start somewhere.

"Waltz, we don't 'have to' start anywhere."

Yes, we do. I want my self-driving car so I can go bar-hopping. And I want it NOW. I'll even accept it before my promised flying car, thanks.

Anyway, getting back to serious. Data. Unlike some people, I can be persuaded by logic. If it turns out that reliable data show that AVs are safer, I'll be convinced.

But there will always be technophobes who will insist that it's better to risk being killed by a human driver than take a lower risk of being killed by a robot driver. Because robots are scary. This is akin to being scared shitless of flying, despite all the statistics. (I despise flying, myself, but that's because they've managed to turn it into the world's most uncomfortable travel experience; I feel fairly safe flying, just grumpy.)

And those are the people we need to convince. Not me. Problem is, I don't know how to appeal to emotion. So meanwhile, I'll just continue to drink at home, and in places where Uber can happen.

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