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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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May 22, 2021 at 12:02am
May 22, 2021 at 12:02am
#1010546
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PROMPT May 22nd

Write about the biggest challenge you have faced and how you overcame it.

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Superlatives can be tricky.

Most embarrassing moment? Greatest fear overcome? Worst mistake? Biggest challenge? All of these things depend on elapsed time since the incident, current situation, and a memory that's inherently faulty.

It's entirely possible that my biggest challenge was learning to walk, a thing that almost all of us have accomplished, but I have only the vaguest memories of that period of my life. Or perhaps it was the opposite: my parents, like I suspect most parents, spent the first three years of my life teaching me to walk and talk, and the next 15 telling me to sit down and shut up. Doing so is a challenge for a kid.

Some time after that, though, I went to college. Now, as you know, I'm a supergenius, so you'd think college would have been easy for me. It was not. Intelligence (I might be overstating mine to some degree) isn't correlated with the drive to do the necessary work to develop it. In other words, I was too busy playing video games learning to drink (not simultaneously; I still suck at that combined skill) to concentrate on coursework. I passed. Eventually. Just barely.

But I did learn one important thing, which was how to focus on learning stuff so I could pass exams. One might argue that this is a relatively unnecessary skill after college, but this is not the case for engineers. So several years later, I had to take the professional engineering (PE) exam, which is not an easy test. Well, I suppose it is for some people, but given a choice between anything resembling work, and playing a video game, I'll play a video game because I'm inherently lazy. Okay, I didn't "have to" take the exam, but having that certification meant more money which I could then use to buy video games and booze.

So when it came time to schedule my PE exam, I buckled down and (mostly) resisted the urge to play video games or drink beer when I should be studying. This gave me skills that the social atmosphere of university never did, which was to find ways to enjoy learning, and to focus on a goal, not my immediate satisfaction. And so I did, and I passed the exam on the first try. To this day, I enjoy learning almost as much as I enjoy those other things. Giving up instant gratification for future rewards, well, I'm still not very good at that.

And then, once it was over, but before I found out the results, I spent a week doing nothing but playing video games in my spare time. Not drinking. Like I said, for me the two are mutually exclusive.

Right now, I'd call that my biggest challenge thus far. Ask me tomorrow, though, and I'd probably pick "losing weight" or "putting the video game on pause long enough to do my taxes." I don't know; I'm not big on remembering the challenge, only the results.

It's also entirely possible that, in the right mood, I would admit that my greatest challenges were ones that I did not end up overcoming. But I'm not in that mood right now. Maybe another time.


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