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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

September 14, 2022 at 12:05am

#1037685

Gut Wrenching

Just found this one today, and it already showed up. This means I might actually remember what I was going to say about it.

Want to be happy? Don’t follow your gut.
A data scientist on what truly makes us happy.

I know I've gone on about happiness in here before. In short, I think there are more worthwhile pursuits, and some of them actually can lead to happiness.

Should you trust your gut?

Gut, heart, brain, gonads. Whatever. I trust my liver.

We’re making all kinds of decisions every day. Most of them are trivial, like what to cook for dinner. Some of them are monumental, like whether to change jobs or sell your house.

We probably only have the illusion of making decisions, but again... whatever.

But every time we make these decisions, we make them on the basis of some feeling or evidence. Sometimes we just go with our intuition, with what feels right. And sometimes we lean on our reason. We weigh the options, consider all the factors, and follow the logic wherever it leads.

Somehow I don't think "logic" figures into it for many people.

A new book by Seth Stephens-Davidowitz, called Don’t Trust Your Gut, argues that our “gut” — or whatever you want to call it — is usually wrong.

I'll buy that. The argument, that is; not shelling out for the book.

And it’s wrong because our intuitions are often influenced by false impressions or dubious conventional wisdom.

"Conventional wisdom" sounds an awful lot like "common sense," which, as I've noted repeatedly, is neither.

Stephens-Davidowitz is an economist and a former Google data scientist.

I give economists a hard time, I know, but their thinking is surprisingly applicable to situations other than financial. Which doesn't mean they're right.

So I invited him to join an episode of Vox Conversations to talk about it.

I don't do podcasts, and I'm not about to start now. It's at the link if you're interested. Or at least I assume it is; my script blocker keeps me from seeing much more than the text. So I'm going by the excerpt provided; the rest of the article is in interview format. Thus, it's hard to quote much of it.

I always feel like if our lives are inefficient enough, you can make decisions that win on every possible dimension.

Now, that's a statement I need clarification on. But the article doesn't provide it.

So I talk about the data on happiness, and particularly the Mappiness Project, by George MacKerron and Susana Mourato, where they asked people on their iPhones: Who are you with? What are you doing? And how happy are you, 0 to 100? And they built this chart, a happiness activity chart.

It's notoriously hard to quantify subjective feelings. Chances are, you're familiar with the pain scale at the doctor's. What's your current level of pain, from 1 to 10, with 10 being the worst pain you've ever felt? Someone who's led a sheltered life might call a banged toe a 10. For someone like me, having had appendicitis, back pain, and a heart attack, it's more like a 4. The point here being that just as you can't compare pain between people, you can't compare happiness either. Different things make us happy, and my happiness at, say, drinking a fine Scotch might actually be greater than your happiness at finally getting laid.

Speaking of which:

So the happiest activity, according to Mappiness — and actually every experience sampling project has landed on the same exact finding — is that sex and intimacy and making love are the happiest activity, which is not so surprising.

I've had way better experiences. Not all of them even involve alcohol.

Gardening ranks really high. Theater, dance shows, sports, running, exercise, singing, performing — so karaoke, really good — talking, chatting, socializing, bird-watching, nature-watching, walking, hiking, hunting, and fishing.

Most of those things are mildly pleasurable for me at best and, at worst (sports, hunting, fishing, e.g.) are things I actively hate. Some are just boring.

You know, another interesting thing about a lot of those activities near the top is that they don’t require a lot of money. A lot of them you can do for free.

Not sex, though. But here's the money question (literally):

Which I guess prompts the question: Did you find that having more money makes us happier? Do you find that happiness tends to scale with income?

Anyone who believes that the best things in life are free has never had a really good bottle of single-malt scotch.

There’s this famous idea that once you get above $75,000 in income, there’s no gain to money. That is kind of a famous idea: You just need at least $75,000 income, then it stops.

Matthew Killingsworth at UPenn did a study. He found that’s not true, that there’s no point at which money stops giving people happiness. But it levels off. So it’s always going up, but it’s going up at a smaller and smaller rate.

Yeah, that $75K idea never really sat well with me. But I can believe the "smaller and smaller rate" thing. It's a principle from economics (unsurprisingly, considering the source) called marginal benefit. Or something like that.

There’s another study by four professors, most of them at the Harvard Business School, that found that there’s an additional boost if your net worth gets above $8 million. And I think one of the reasons for that is, if you think of the activities that are really at the bottom of the happiness activity chart, there are these annoying things that modern life forces us to do. And once your net worth gets to $8 million, you really can stop doing them. So you just have a chef cooking your meals and you have a housekeeper who’s cleaning up after everything. And maybe you have a personal assistant who’s doing all your chores, and you have a personal driver, so you’re not commuting on a subway.

Sorry, guys. Rich people really are happier, and they get their happiness by using the rest of us. (You think the rich bastard's housekeeper is swimming in serotonin?)

Well, the trap is that work is the second most miserable activity according to [scholars Alex Bryson] and MacKerron, which shocked me because I had grown up with this idea that work is where you get a lot of your fulfillment and joy and purpose.

Yeah, that's another lie they tell us along with "money can't buy happiness" and "the best things in life are free." It's designed to keep us from wanting more.

You have a chapter in there about parenting and kids. What did you find that makes a good parent? What did the data tell us about how to parent better?

I've heard a lot of people say they derive their happiness from having kids. I'm sure there are some people like that; I knew from an early age I wouldn't be one of them. True happiness, for me, requires that I not have too many obligations, and kids are a huge obligation.

They also go on to talk about happiness in marriage, and the bit I find interesting there is this:

The thing that predicts happiness — by far the most important predictor of whether you’re happy in your romantic relationship — is whether you’re happy outside your relationship.

This is not what I'd consider new information, but it's quite telling that the data seem to back it up: that if you're not content in yourself, you're not going to find contentment with someone else. People searching for a "relationship" in order to be happy are looking in the wrong place. Which is not to say people can't be happy in a relationship; just that you gotta work on yourself first.

I would say that the most depressing finding in the book is probably also the least surprising, which is that basically being good-looking is the most predictable determinant of success in almost every sphere of life.

It's only "depressing" if you're already ugly. We knew this, too, though; rich and attractive people generally seem happier than poor ugly slobs.

To the extent that I sounded a skeptical note, part of what I was getting at is, I just feel like human beings are just sort of hopelessly contradictory. You know, like if every day was a perfect day, then pretty soon the things that made it perfect would cease to satisfy us. Right? You know, the sweet is only sweet because of the sour, and all that.

Both the people involved in this interview admitted to having been philosophy majors in college. Nothing wrong with that, but that bit of philosophy right there is an echo of what the great philosophers Beavis and Butt-Head once said: "If everything was cool, and nothing sucked... how would we know what was cool?"

In the end, I think the data here are important, but they shouldn't guide anyone. It's like... statistics show that there are more cats than dogs kept as pets in the US (though more households have dogs, there are more cats per household, on average). From that data alone, you might draw the conclusion "Cats are better pets than dogs. Therefore, I should adopt cats." Now, I happen to agree with that—but I have a very good friend who's more of a dog person, and she wouldn't agree. And that's okay. All I'm saying is, just because the data says, on average, "you'll be happier if you took a long walk in nature and end up doinking your romantic partner on a lakeshore" (which apparently it does), that doesn't mean that'll do it for all of us. Especially indoor-dwelling asexuals.

Not to mention the findings about work. I know a few people for whom work gives their life meaning, and meaning leads to some happiness. Again, not me, but that's my point: we're all different, and we all enjoy different things to different degrees.

Despite my objections, though, I can't say the discussion was uninteresting. Though still not interesting enough to listen to the 'cast or buy the book. Just interesting enough to produce enough food for thought to create a blog entry. So... here it is.

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