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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.




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August 10, 2024 at 10:44am
August 10, 2024 at 10:44am
#1075011
Today marks one month since I first talked to the travel agent about my upcoming trip. This week, they sent me a proposed itinerary, with the understanding that I could make changes if I wanted.

But there's no need for that. Well, maybe some minor tweaks, but what I saw was exactly in line with what we'd discussed last month. It's almost as if they listened to me and found ways to incorporate what I said I wanted out of the trip. Two weeks in France, one week in Belgium. Food, wine and beer. Some guided tours. Some unguided time.

So I wrote back with a request for a small change, basically just a visit to a particular Belgian brewery. Not even an abbey; just a regular business (one that happens to be almost as old as my entire country).

There's only one frustrating thing about this, which is that I still don't have an actual schedule. Their proposal was like Day 1, Day 2, etc., but nothing about dates of travel. As I expected to leave in the latter part of September, it seems like that's something we should nail down. But if the trip's in October, that's fine. Just not November. Too cold.

And I really should get some travel gear together, but apparently, I'd rather procrastinate.


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