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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 4 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

3 Like

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.

2 Like

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.

1 Like

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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

August 27, 2022 at 12:02am

#1036983

Lucky Strikes

Today's your lucky day. Oh, wait... no. Well, maybe it is. But I'm here to talk about luck.

The radical moral implications of luck in human life
Acknowledging the role of luck is the secular equivalent of religious awakening.

I think the headline goes too far, bordering on clickbait. But that doesn't mean the content is wrong.

In July 2018 (when we first published this piece), there was a minor uproar when Kardashian scion Kylie Jenner, who is all of 22, appeared on the cover of Forbes’s 60 richest self-made women issue. As many people pointed out, Jenner’s success would have been impossible if she hadn’t been born white, healthy, rich, and famous.

People say the Kardashians are famous for being famous. I don't know. I don't keep up with them (pun intended). I don't care about them. Their antics mean exactly nothing to me. I'm rather disappointed that Forbes, a magazine that my very serious father always read very seriously, would fall so low as to cover one of them for any reason, but hey, publications change. But anyway, no, they're not famous for being famous; they've invested a lot of time and money into self-promotion, netting them more money (if not more time). I can't help hearing about them, no matter how strenuously I try not to. I get the feeling that if I moved out to the middle of Nevada and started living under a rock in an abandoned silver mine, within a week someone would come trudging into the mine, lift the rock and go, "Hey, did you hear about the latest thing some Kardashian did?"

All of which is to say that it's not just luck. Or if it is, it's lucky that they're so talented at self-promotion.

She built a successful cosmetics company — now valued at $900 million, according to Forbes — not just with hard work but on a towering foundation of good luck.

"Now" apparently being the date of the updated article, early 2020, still in the Before Time.

And I will reiterate that the idea that "hard work" (whatever that is) alone brings success is easily refuted: if it were the case, migrant laborers would be billionaires.

Around the same time, there was another minor uproar when Refinery29 published “A Week in New York City on $25/Hour,” an online diary by someone whose rent and bills are paid for by her parents.

A publication I've never heard of. But I've seen similar articles, usually with the tone of "This young couple managed to buy a house and pay it off in full by the time they were 30," while if you actually read the article, you find that they were able to do so because their parents paid for most of their crap.

It’s not difficult to see why many people take offense when reminded of their luck, especially those who have received the most. Allowing for luck can dent our self-conception. It can diminish our sense of control. It opens up all kinds of uncomfortable questions about obligations to other, less fortunate people.

Oh, I don't take offense. I just smile (insofar as I can), narrow my eyes, and go, "So?"

Nonetheless, this is a battle that cannot be bypassed. There can be no ceasefire.

And now we're back to the hyperbolic tone of the headline.

Individually, coming to terms with luck is the secular equivalent of religious awakening, the first step in building any coherent universalist moral perspective... Building a more compassionate society means reminding ourselves of luck, and of the gratitude and obligations it entails, against inevitable resistance.

I find that to be somewhat contradictory. If I'm playing craps and I make my point, and I'm an atheist (remember, they're talking about secular morality here), to whom or what do I express gratitude? The dice? That's silly. God? Nonexistent. Lady Luck? Still atheist. Fortuna, Roman goddess of luck? Still atheist. Some nebulous concept of the quantum fluctuations of the universe? See "dice." Sure, if an actual person does something nice for you, you express gratitude. Sure, gratitude is actually an emotional state and is intransitive (meaning it doesn't require an object), but if I win at gambling, I don't say "thank you;" I just bask in the fortune.

All of which is to say I'm not convinced that "gratitude and obligations" are a necessary byproduct of being lucky. But I'm willing to read on to see what the author might have to say about that.

How much moral credit are we due for where we end up in life, and for who we end up? Conversely, how much responsibility or blame do we deserve?...

How you answer these questions reveals a great deal about your moral worldview. To a first approximation, the more credit/responsibility you believe we are due, the more you will be inclined to accept default (often cruel and inequitable) social and economic outcomes. People basically get what they deserve.

I think those are fair questions, and I accept that people will answer them differently. In my worldview, we only have the illusion of being able to make decisions. It's more like we do whatever it is that we do, and then either justify or regret it afterward. It's also very clear to me that people do not, in general, get what they deserve; it's more like they get something, and have to (or get to) live with it.

The idea that people get what they deserve is pernicious. You end up worshiping successful people, and scorning those in poverty, on the basis of "well, they must have done something to deserve their state." (The article does delve into this morass later.)

Of course it is true that you have no choice when it comes to your genes, your hair color, your basic body shape and appearance, your vulnerability to certain diseases. You’re stuck with what nature gives you — and it does not distribute its blessings equitably or according to merit.

But you also have no choice when it comes to the vast bulk of the nurture that matters.

On that point, I can agree. You can no more choose your parents, or your childhood environment, than you can choose your eye color (please don't tell me about colored contact lenses; you know what I mean).

Here, a distinction made famous by psychologist Daniel Kahneman in his seminal Thinking, Fast and Slow is helpful. Kahneman argues that humans have two modes of thinking: “system one,” which is fast, instinctual, automatic, and often unconscious, and “system two,” which is slower, more deliberative, and emotionally “cooler” (generally traced to the prefrontal cortex).

Our system one reactions are largely hardwired by the time we become adults. But what about system two?

We do seem to have some control over it. We can use it, to some extent, to shape, channel, or even change our system one reactions over time — to change ourselves.

The key word there, to me, is "seem." This argument implies that we are somehow separate from our selves, that there's a ghost in the machine, pulling the levers, and we're the ghost. The problem with that implication is that, well, we're not. System one, system two, whatever; they're both products of brain activity—products of a physical process.

We do change, sure. Other environmental inputs give us more information, and the brain itself changes over time.

Everyone is familiar with that struggle; indeed, the battle between systems one and two tends to be the central drama in most human lives. When we step back and reflect, we know we need to exercise more and eat less, to be more generous and less grumpy, to manage time better and be more productive. System two recognizes those as the right decisions; they make sense; the numbers work out.

But then the moment comes and we’re sitting on the couch and system one feels very strongly that it doesn’t want to put on running shoes. It wants greasy takeout food. It wants to snap at the delivery guy for being late. Where is system two when it’s needed? It shows up later, full of regret and self-recrimination. Thanks a lot, system two.

This is usually represented in cartoons with an angel on one shoulder and a devil on the other.

As an aside, I take issue with the idea that we should be more productive. Productivity has led to myriad problems. Maybe we could do with being less productive.

I'm skipping a bunch here, but something else I have issue with:

The promise of great financial reward spurs risk-taking, market competition, and innovation. Markets, properly regulated, are a socially healthy form of gambling.

No. They are not gambling. I mean, sure, if you take a short-term view, they can be. But unlike, say, gambling in a casino, investment in the stock market gives you the house edge. So unless you're also prepared to call running a small business or a casino "gambling," this is another pernicious misconception.

And there’s no reason we shouldn’t ask everyone, especially those who have benefited most from luck — from being born a certain place, a certain color, to certain people in a certain economic bracket, sent to certain schools, introduced to certain people — to chip in to help those upon whom life’s lottery bestowed fewer gifts.

Oh, you can ask all you want. You can even enforce the ask with taxes. But those types are lucky enough to be able to afford lawyers and accountants to minimize their tax burden. In the end, it doesn't matter whether someone knows they were lucky or not; what matters is how much they give a shit.

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