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

June 12, 2022 at 12:15am

#1033738

In Pursuit

Here we go again with "happiness." It's enough to make me grumpy.

10 Practical Ways to Improve Happiness
For when you need advice that goes beyond “Be Danish”

How about "Eat a Danish?"

Here’s some very bad happiness advice based on very solid happiness research: Feel important. Be happily married. Be Danish.

Lots of people feel important and yet are angry as hell. Just walk into a store or restaurant and you'll see what I mean.

Maybe become a Great Dane, instead? Dogs seem happy. Too happy, if you ask me.

Based on what they see in the data from experiments and surveys, what should we do that is both effective and feasible for increasing our happiness, starting today?

I've said it before, and I'll say it again: I don't believe happiness is, or should be, a goal. It's a byproduct of doing other stuff. Like, I don't drink to be happy, but when I drink, I become happy.

Here are the top 10, in order, with my own assessments as a happiness researcher added in for good measure.

Oh boy oh boy this is gonna be fun!

1. Invest in family and friends. The research is clear that though our natural impulse may be to buy stuff, we should invest instead in improving our closest relationships by sharing experiences and freeing up time to spend together.

Okay, but "stuff" won't let you down. Oh, sure, it eventually wears out or breaks, but it doesn't deliberately ghost you like friends and family do. And then it's easier to replace.

2. Join a club. The “social capital” you get from voluntarily and regularly associating with other people, whether or not you do so through a formal club, has long been known to foster a sense of belonging and protect against loneliness and isolation.

"I refuse to join any club that would have me as a member." -Groucho

3. Be active both mentally and physically. You can make this advice as complicated and expensive as you want. But if you like to keep things simple, just try to walk for an hour and read for an hour (not for work!) each day.

Yeah, right, because... no, actually, I can't snark on this one.

4. Practice your religion.

And this is where I'd stop reading if I weren't dedicated to making fun of shit.

5. Get physical exercise. This is a slightly souped-up version of No. 3 above: Your daily walk should be supplemented with a purposive exercise plan.

Now you're just talking crazy talk. And making up words. (To be fair, I make up words too.)

6. Act nicely. Agreeableness is consistently found to be highly and positively correlated with happiness, and it can be increased relatively easily.

Oh, sure. Yep. Act nice. Be agreeable. Don't stand up for yourself. Don't argue. Plaster a fake smile on your face, and fake it til you make it.

Except bullshit. While I don't think we should go out of our way to be mean or nasty to people, there are certainly times when "be nice" would increase your anger level and decrease your happiness.

7. Be generous. Behaving altruistically toward others rewards the brain with happiness-enhancing boosts of dopamine, serotonin, and oxytocin.

Until they start expecting it, when they become mooches.

8. Check your health. Of all health issues, those that create the greatest unhappiness are typically chronic pain and anxiety.

Oh, wow, I'd never have guessed that pain and anxiety might be barriers to happiness. Incredible. I guess it takes a degree in happiness science to figure that one out.

9. Experience nature. Studies have shown that, compared with urban walking, walking in a woodland setting more dramatically lowers stress, increases positive mood, and enhances working memory.

"That's the problem with nature. Something's always stinging you or oozing mucus on you." -Calvin

To be clear, I do love nature, and I very much enjoy watching it from a temperature-controlled, sealed environment. Although I still maintain that the distinction between "nature" and "artifice" is entirely... well... artificial.

10. Socialize with colleagues outside of work. Data have shown that work friendships increase employee engagement, which is associated with both happiness and productivity for workers.

What the... NO. Leaving aside for a moment that prayer to Holy Productivity, who wrote this, a corporate executive?

So here's my version. You knew this was coming, didn't you?

1. Don't have goals. You will only fail them and that will make you unhappy.
2. Waste time at work.
3. Drink.
4. Expect the worst so you can only be pleasantly surprised.
5. Cats.
6. Read (or watch or listen to) comedy.
7. Music.
8. Minimize the amount to which you are dependent upon other people.
9. Stand up for yourself (i.e. don't always be nice, accommodating, or agreeable).
10. Snark on vapid internet articles about happiness.
11. (Bonus!) Drink some more.

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