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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December 8, 2022 at 12:02am

#1041541

Think Tanked

Today, we're going to think about thinking.

Twenty-Five Useful Thinking Tools

There's a word for that: Metacognition. Which, incidentally, would make an awesome name for a steampunk Metallica cover band.

Most people think being smart is about having more facts.

I know I've noted this before, and this statement (and its refutation) is one reason I saved this article to my queue.

This is probably the least important and useful part of learning though. Instead of facts, I’d prefer to focus on knowledge that acts as tools. The more you have, the more ways you can approach different problems.

I don't know about "least important," however. It's certainly helpful to have facts at the tip of one's metaphorical fingers, or at least to know how to find them quickly. For example, I couldn't remember the word "metacognition," but I knew it existed, so I Googled "thinking about thinking."

The thing about facts, though, is that not everything is either "fact" or "not fact." For example: The Dallas Cowboys won the 1978 S****wl. This is a fact; there is no disputing it. Denver fans may have been butthurt about it, but they had to acknowledge the result. (The same is true for the last US Presidential election, like it or not.)

This, however, is trivia.

Other things are plainly NOT facts, regardless of whether you want them to be. Like "the moon is made of green cheese," or "Earth is flat."

Some things that we think of as "facts" are really "things we know to varying degrees of certainty." The relative abundance of hydrogen and helium in the Sun, for example. Or that the universe has existed for approximately 13.7 billion years. New discoveries may change or refine these numbers.

The point is, what we want to do is to be able to tell facts from not-facts, not to memorize trivia. Unless you're trying to win trivia contests, which admittedly is a perfectly acceptable goal—but it wouldn't mean you're smarter than the other contestants.

Charlie Munger, Warren Buffet’s long-time investing partner at Berkshire Hathaway, calls these mental models. More mental models means you have more ways to solve more problems.

In my view, how you think is more important than what you think. Unless you think the Earth is flat, in which case, boy are you in the wrong place right now.

Professions as Thinking Toolkits

Most people define professions by what those professions do. Engineers build things. Economists study money. Psychologists look into people’s minds.

However, while this is an obvious distinction, I’m more interested not in what types of problems professions try to solve, but how they try to solve them. Here, we can uncover a wealth of different thinking tools that are often abstract enough to apply well outside the typical interest of the profession.

I think (there I go again) this is an excellent way to look at various mental models, or ways of thinking about things. There are, of course, other ways; that's the whole point. But I linked this article because I like the paradigm. Which is not to say that I completely agree. Nor am I going to list all the different viewpoints here; that's what the link is for.

I will note this one, because it's of personal interest to me:

3. Engineer: Can I Model This and Calculate?

Engineering, being built off of the hard sciences, has some of the most precise and accurate estimates in any profession.

That's a stretch, especially in my own field of civil engineering. We overdesign things on purpose to account for unknowns. It's not rocket science, which famously does require precision and accuracy.

The rest of this section, though, is pretty much on the ball.

Also, this one:

6. Journalist: Just the Facts

Journalists rely on a ton of different thinking tools which allow them to write compelling stories that report the news fairly and accurately.

One of these thinking tools is fact-checking. Because journalists often need to interview sources who may be misleading (or even hostile), it’s important to corroborate what was said from independent sources. Fact-checking may be time consuming, but it results in a much more accurate world-view than simply blindly following a stray comment.

Now, sure, we can make comments on the lack of application of this in the real world, but ideally, yes. That aside, I'm highlighting this one because of the last time I noted the difference between knowing trivia and being smart.

That was in this entry: "Know One"

(yes, I slogged through old posts to find it; turns out it was almost exactly one year ago). In it, I wrote:

I watched a thoroughly mediocre science fiction show on Amazon a while back. As part of the plot, a journalist and a scientist walk into a bar (yes, that could be a setup for a joke). The bar is having Trivia Night. The scientist is portrayed as absolutely confident that he will win the trivia contest, and he and the journalist bet each other.

Thing is, scientists are 1) not automatically more intelligent than anyone else; 2) generally focused on one or two narrow fields of study and 3) entirely too immersed in said study to know, or care, who played Steve on Soaps of our Lives or who was the second-string wide receiver for the Raiders in 1997. The show's writers fell into a common trap: conflating intelligence with knowledge.

All else being equal, I'd put all the money on the journalist in a trivia contest against a scientist.

Point being that journalists, in general, will have broader knowledge, while scientists tend to have deeper knowledge. The two aren't incompatible, however. And as this linked article points out, neither is better; they just come from different thought models.

Ironically (or coincidentally or whatever), the article's very next section is:

7. Scientist: Make a Hypothesis and Test It

A basic thinking tool of science is the controlled experiment. Keep all the variables the same, except the one you want to test, and see what happens. This requires meticulous preparation and design to prevent outside contamination from breaking your results.

Too many people draw inferences from “experiments” that are anything but.

Say you have a cold. You take a couple of Snake Oil pills for it. Your cold clears up. Natural conclusion: "Snake Oil pills cured my cold!" Well... no. I mean, maybe. But maybe it would have cleared up at that point anyway. There's an old joke that goes something like: have a cold, do nothing, it lasts 14 days. Have a cold, take medicine, it lasts a fortnight. I don't remember the exact phrasing and I'm running out of time here to look it up, but the point is a study with one data point solves nothing; it could very well be coincidence.

Anyway. Like I said, running out of time here. I think the important takeaway from the article is that, regardless of what you do for vocation or avocation, you can benefit from using some of these thinking tools.

I'll include just one more, because it's relevant to most of those reading this:

22. Novelist: Does Your Story Make Sense?

Novelists understand better than anyone that what actually happens is often not a good story. Stories have characters with fixed traits that make their actions predictable. In real life, people are more influenced by context. Stories have beginnings, middles and ends. Reality is a continuous stream of events without an arc.

Unfortunately, people understand stories much better than realities. So often you need to package up the histories you want to tell people in a way that they can interpret. Who is involved? When did those things happen? Give information to make it easier for the listener to follow.

While this applies to writing novels or making movies, telling stories is a part of everyone’s life. From “Why do you want to work at this job,” to, “Where do you see yourself in five years?” These are all stories, and we need to understand their structures.

Which only reinforces another thing I've been saying: There's no such thing as useless knowledge. And now I can extend that to: There's no such thing as useless ways of thinking.

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