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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.
November 21, 2023 at 9:35am November 21, 2023 at 9:35am
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Three randoms about luck in a row. Must be a lucky streak.
This one's from GQ, and I have some issues. Wait, no, I don't have any issues of GQ. I mean I have complaints.
Some guys seem to have all the luck. A perfect career, a perfect partner, a perfect life.
"Seem to have" is the key phrase here. You never know what inner battles he's fighting. (I excuse the gendered language because the outlet's target audience is men. I think they switched to GQ because it's no longer quarterly, not because "gentleman" is offensive. But I'm not sure.)
When they’re not sitting next to a book publisher on a flight, they’re discovering a vintage Burberry trench in the thrift store around the corner from your apartment. It’s unbelievable. It’s annoying.
The only thing annoying here is "your" jealousy and "your" need to compare yourself to others.
Their luck seems random—and these days, thanks to social media, it seems like everybody’s getting lucky but you.
Again, "seems" is key. In both parts of the sentence. The article is from 2018, but social media is still here, and on it, there's a selection bias whereby you mostly see the awesome stuff (unless they have a GoFundMe they want you to contribute to).
That’s because luck isn’t something that happens to you; it’s something that happens because of you.
And this is where I lose my shit.
"Why? Isn't that affirming? Isn't that good news to us unlucky schlubs who can't seem to catch a break, to know that we actually have the agency to become lucky?"
No.
Because 1) it's not true and 2) the corollary to "luck is something that happens because of you" is "if bad things happen to you, it's your fault."
Say, for instance, your spouse cheats on you. What would you rather hear? "That sucks," or "You could have been a better partner?"
Your leg gets run over by a truck. Wrong place, wrong time? Or was it your fault for not choosing a better place to be?
Mind you, I'm not saying we can't stack the deck. You can be pretty sure you won't get killed by a shark if you never go near the water (your chances of being killed by a shark on land are low, but never zero). Hygiene and grooming go a long way when it comes to making an impression on others. And you can certainly plan for certain eventualities, like being sure to have roadside assistance lined up before going on a road trip, just in case.
In other words, your house burning down is bad luck. You having insurance so you can rebuild is planning... but even that requires that you be lucky enough to live in a place that offers that, have the means to pay for it, and have it available.
Let's look at one more passage from the intro:
Make a few tweaks to the way you approach opportunities that arise in your daily life and you too can become one of the savvy and brave people capable of making their own lucky breaks happen.
And then, if you don't get lucky, we all know you just didn't try hard enough.
From the interview portion of the article:
GQ: You’ve written that there is a “physics” to luck, since all of life is a matter of cause and effect. What do you mean by that?
Tina Seelig: We live in a world where every single choice you make has consequences.
Invoking physics into this is an insult to actual physics.
So what are those behaviors you can practice to attract luck?
One is showing appreciation.
That's not attracting luck; that's getting on others' good sides.
The other is taking risks. Go up and say hello to somebody you don’t know...
One wonders if this interviewee has ever been approached by someone who doesn't know her, say, on the NYC subway.
And finally:
Right. I also think it’s really important to distinguish between fortune, chance and luck. People don’t distinguish between them.
That's because they're actually synonyms, though it's true they can have different connotations. For example, "a fortune" is not the same thing as "fortune," though most people with a fortune are considered fortunate, but you don't have to be rich to be fortunate, and someone reading your fortune might note that you'll never amass one.
Fortune is things that are outside of your control, things that happen to you.
No.
That is what luck is.
Chance is something you have to do; I have to take a chance.
No.
Chance is accident. You find a $20 on the sidewalk by chance (though you did prepare for that by looking down instead of up). You run into an old friend at a bar in another town by chance.
Luck is something where you have even more agency. You make your own luck by identifying and developing opportunities in advance.
And absolutely, positively, NO.
So, okay, is my basic objection to the thrust of this article based merely on a semantic argument?
No, I don't think so. However you want to define your terms, there are some things we simply don't have agency over. While it's probably true that you won't win a lottery jackpot if you don't buy a ticket (as with shark killings on land, chances are low but never zero), if you do possess a ticket, you have absolutely no control over the winning numbers. You have absolutely no control over the genetic hand you've been dealt. You have absolutely no control over what your early upbringing was like. You *do* have at least the illusion of control over how you play your cards, and that can make a big difference in your life.
Let me close with an example from popular culture: Willy Wonka and the Chocolate Factory. Of course it's a story and it proceeded as the writers wished, but consider it a parable of sorts.
As you know, Wonka hid Golden Tickets in candy, the winner of which would get a tour. Now, there's some debate over whether the tickets were truly distributed randomly, or, like McDonald's Monopoly pieces, predestined. But everyone assumed it was random and acted as such, so from their point of view, it might as well have been.
I'll just focus on two of the approaches used to acquire the coveted Golden Ticket: First, Veruca Salt, who demanded that her father buy every available Wonka bar and have his horde of employees peel them looking for the ticket. This is the "hard work" mentality, the idea that the more you do, the better your chance of reward. Sure, Salt had minions doing the actual work, but the result is the same.
This is, absolutely, a valid way to go about it from a mathematical (if not humanitarian) perspective. The more candy bars you buy, the better your odds of winning the prize.
Of course, you're not supposed to like Veruca Salt, because she's a spoiled brat. (I met Julie Dawn Cole a few years ago, though, and she was a thorougly pleasant person.)
You are supposed to like Charlie Bucket, who lacks the resources of Salt and the work ethic of her father's wage slaves. And because Charlie's featured in the movie, you know he's going to end up with a ticket, but his approach is: buy the one bar you can afford, and hope for the best.
Thing is, moral lessons aside, both methods—hard work, exemplified by the army of minions peeling bars, and blind luck, exemplified by Charlie's fortunate find—worked equally well. And yet, we're supposed to hate the one who did the hard work (okay, she made others do the hard work, but it was still hard work) and root for the one who just got lucky. Which is utterly backwards from the narrative we usually get fed.
Even the hard work involved an element of luck, because unless she purchased every single chocolate bar, there was never a guarantee, only better odds. Higher probability. Not to mention she already got lucky by being born into a wealthy family.
Didn't stop her from getting incinerated, though. And that bit really was her own fault. |
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