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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July 25, 2022 at 12:01am

#1035662

The Disconnected You

It is fitting that I'm just now getting around to sharing today's article, which is from 2014.

Why We Procrastinate
We think of our future selves as strangers.

Can I just say how refreshing it is to have the answer to the implied headline question right there in the subhead? Neat, short, concise even.

Of course, the article delves into much more detail.

The British philosopher Derek Parfit espoused a severely reductionist view of personal identity in his seminal book, Reasons and Persons: It does not exist, at least not in the way we usually consider it. We humans, Parfit argued, are not a consistent identity moving through time, but a chain of successive selves, each tangentially linked to, and yet distinct from, the previous and subsequent ones.

You know I find philosophy intriguing. But this? This sounds like a modern-day version of one of Zeno's Paradoxes.

Parfit’s view was controversial even among philosophers.

I'm sure it was. Not that it's definitely wrong, mind you; it's just that I can't think of a way to test it. Therefore, it stays in the realm of speculation—just another way of looking at things.

But psychologists are beginning to understand that it may accurately describe our attitudes towards our own decision-making: It turns out that we see our future selves as strangers.

Your future self is entirely theoretical. Your past self is not (except in the view of fringe philosophers). There's a big difference there.

That bright, shiny New Year’s resolution? If you feel perfectly justified in breaking it, it may be because it feels like it was a promise someone else made.

That's a non sequitur if I've ever seen one. In that case, you're breaking a promise your past self (aka you) made. Now, whether your past self has any business dictating stuff to your future self, well, that's another issue entirely.

Using fMRI, Hershfield and colleagues studied brain activity changes when people imagine their future and consider their present. They homed in on two areas of the brain called the medial prefrontal cortex and the rostral anterior cingulate cortex, which are more active when a subject thinks about himself than when he thinks of someone else. They found these same areas were more strongly activated when subjects thought of themselves today, than of themselves in the future. Their future self “felt” like somebody else. In fact, their neural activity when they described themselves in a decade was similar to that when they described Matt Damon or Natalie Portman. And subjects whose brain activity changed the most when they spoke about their future selves were the least likely to favor large long-term financial gains over small immediate ones.

I'd definitely keep a promise I made to Natalie Portman. Just saying.

The disconnect between our present and time-shifted selves has real implications for how we make decisions. We might choose to procrastinate, and let some other version of our self deal with problems or chores.

Well, absolutely. I don't want to deal with them. Let someone else do it. Sure, that someone else is future-me, but hey, I might get lucky and die before I have to deal with them.

Anne Wilson, a psychologist at Wilfrid Laurier University in Canada, has manipulated people’s perception of time by presenting participants with timelines scaled to make an upcoming event, such as a paper due date, seem either very close or far off. “Using a longer timeline makes people feel more connected to their future selves,” says Wilson. That, in turn, spurred students to finish their assignment earlier, saving their end-of-semester self the stress of banging it out at the last minute.

Wait, what? Some people don't get stuff done at the last minute?

Of course, the way we treat our future self is not necessarily negative: Since we think of our future self as someone else, our own decision making reflects how we treat other people.

"Other people?" You mean the game's NPCs?

So anyway, yeah, short article, interesting implications (if true). Eventually you might even get around to reading it. Not sure it has much to do directly with the Parfit philosophy from the first quote up there, but I suppose it fits that model.

Meanwhile, I'll just give myself a pat on the back for finishing writing this at 11:30 instead of 11:59.

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