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

August 23, 2022 at 12:01am

#1036837

Nonsense

To be clear, I started saying "common sense is neither" before I found this article. But then I found this article, so of course I had to put it in my queue. It's from 2011, just to put some of its content into context.

Common Sense Is Neither Common nor Sense
How often is common sense correct?

I started getting the idea that "common sense" was just another phrase for anti-intellectualism and know-nothingness. Some politicians run on a "common sense" platform, and the ones that do are all a bunch of down-home anti-book-learnin' ideologues, so I don't trust them. "Let's ignore scientific evidence and instead just go on my personal experience."

Common sense, defined as "sound judgment derived from experience rather than study," is one of the most revered qualities in America.

That's a silly damn definition. It's not sound, it's not judgment, and if it's derived from experience, then it's not common, is it? It's personal.

It evokes images of early and simpler times in which industrious men and women built our country into what it is today.

Let's be real here. Most "common-sense" folks are only thinking about the men from history. They only care about women in terms of how many babies they can make.

People with common sense are seen as reasonable, down to earth, reliable, and practical.

Nice passive voice there. Not by me.

But here's the catch. Common sense is neither common nor sense.

Which is what I've been saying.

If common sense was common, then most people wouldn't make the kinds of decisions they do every day. People wouldn't buy stuff they can't afford. They wouldn't smoke cigarettes or eat junk food. They wouldn't gamble. And if you want to get really specific and timely, politicians wouldn't be tweeting pictures of their private parts to strangers. People wouldn't do the multitude of things that are clearly not good for them.

Okay, whoa, hold the fuck up there.

People do all those things for plenty of reasons, not all of them being that they lack any kind of sense. People buy stuff they can't afford because they're convinced by ad agencies that it will improve their lives more than saving money will. People smoke cigarettes because they're addicted, or because it's genuinely pleasurable. People eat junk food because it's cheap and easy, cheaper than healthy food; I can't reconcile this, at least for poor people, with buying only stuff you can afford. Gambling can be a problem, but for some of us it's just another entertainment expense, equivalent to going to a sportsball game or music concert. We all do things that aren't good for us because, on some level, they are good for us. (The "private parts" thing is a reference to a scandal that was current when the article was written.)

This doesn't change my agreement with the general thrust of the article, but that bit is condescending as hell.

This is the important bit:

And common sense isn't real sense if we define sense as being sound judgment because relying on experience alone doesn't usually offer enough information to draw reliable conclusions. Heck, I think common sense is a contradiction in terms. Real sense can rarely be derived from experience alone because most people's experiences are limited.

Our senses, our lived experiences, tell us the Earth is flat. It looks flat, doesn't it? Especially if you live in Kansas. And that we're at the center of the universe. A person who survives a car accident while not wearing their seat belt might believe that it's safer to not wear a seat belt. The news covers plane crashes religiously, but rarely car wrecks, so you'd think flying would be more dangerous than driving (it is not). You took horse dewormer and got better, so obviously the horse dewormer made you better, right? That's just common sense.

It takes science, research, knowledge, book learnin' to extend our senses, to help us see reality beyond our limited individual experiences. No, science doesn't have all the answers. But it has way more than you do alone. Science tells us the planet's roughly spherical, and how gravity affects orbits, and that the universe doesn't have a "center." Research shows that you're safer with a seat belt than without one, but that doesn't mean you're invulnerable. Studies show that dewormer only cures worms; if you got better from something else while taking it, that was a coincidence.

The word common, by definition, suggests that common sense is held by a large number of people. But the idea that if most people think something makes sense then it must be sound judgment has been disproven time and time again. Further, it is often people who might be accused of not having common sense who prove that what is common sense is not only not sense, but also completely wrong. Plus, common sense is often used by people who don't have the real knowledge, expertise, or direct experience to actually make sound judgments.

"Global warming isn't real! Look at this snowball."

I think we need to jettison this notion of the sanctity of common sense and instead embrace "reasoned sense," that is, sound judgment based on rigorous study of an issue (which also includes direct experience).

No, this doesn't mean going through youtube or social media in search of things that ring your confirmation bias. Yes, I'm aware that this article rings my confirmation bias. No, this is not a contradiction.

A course in scientific thinking and methodology for everyday life should be a requirement for all students. Such proactive education about precise thinking and real sense might reduce the number of truly dunderhead things that subsequent generations will do (the current generations are probably beyond remediation).

Wow, this guy's a tool. Doesn't mean he's wrong, though.

Without being receptive to answers that we may not want to hear, we might as well just ask ourselves what we want to be true and go with that, which is what many people with so-called common sense (most efficient, but often wrong).

Which is what a lot of people seem to do anyway. (This article could have used an editor.)

Let's be realistic. No one likes to see their "theories" disproven.

This is not true. For instance, I have a "theory" (which isn't one in the scientific sense) that technology-using beings, such as us, are extraordinarily rare in the universe, to the point where there's not another one in this galaxy. I'd love for that to be disproven (unless of course it involves their technology blowing up the Earth). And it would be very easy to disprove it.

Anyway. Quibbles aside, the main thrust of the article is something I absolutely agree with.

Which of course makes it suspect. But I can live with that.

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