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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. ![Merit Badge in Quill Award
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| Today's article is five years old. Given the subject matter, it seems appropriate. Why some people are always running late  It’s not always rudeness or scatter-brained behaviour – it can be something much deeper, writes Laura Clarke. I thought I might have linked this one before, but I couldn't find it in a search. I've been at this for a long time and I have a shit memory. We all know that person: there’s the child minder who is always late, the colleague who misses every deadline, even if just by a few hours, the friend you must tell to arrive 30 minutes earlier than she needs to for your lunch reservation. 30 minutes? Amateur. I dated a woman who had to be asked to show up two hours early. Even then, she was often late for the actual time. Hell, one time we left for a trip 28 hours after we were supposed to. She called to change plane reservations (this was effectively pre-internet) three times. We didn't date very long. Bad enough my first wife was two hours late to our wedding. She had an excuse, at least (illness). A look into the psychology of lateness offers a glimpse into a mind that that may be malfunctioning. Seems to be a lot of that going around these days. Perceptions of unpunctual people are almost always negative — even if misguided. They should be negative. However, I'll tentatively accept the article's assertion that it's not always the reasons we think. Some excuses, particularly for acute lateness, are fairly universally accepted —an accident or illness, for example. But others aren’t so easy to swallow. Some late people will pass it off as a symptom of being big-thinking and concerned with loftier matters than time-keeping, as an endearing quirk, a mark of doing one’s best work under pressure, or having the body clock of a night owl rather than a lark. Okay, go right to hell with all those, especially that last bit. Maybe it's why you're late for work -- I almost always was, and for that reason -- but it's definitely not an excuse to miss your 8pm dinner reservation. Being consistently late might not be your fault. It could be your type. The punctually-challenged often share personality characteristics such as optimism, low levels of self-control, anxiety, or a penchant for thrill-seeking, experts say. Personality differences could also dictate how we experience the passing of time. Well, we can get all philosophical about whether those things are your fault or not, delving into questions of free will, but either way, it's still your responsibility. It's like... maybe you have the kind of personality that makes you love driving fast. I can understand that. But it's still your responsibility to pay the inevitable speeding ticket. In 2001, Jeff Conte, a psychology professor at San Diego State University ran a study in which he separated participants into Type A people (ambitious, competitive) and Type B (creative, reflective, explorative). He asked them to judge, without clocks, how long it took for one minute to elapse. Type A people felt a minute had gone by when roughly 58 seconds had passed. Type B participants felt a minute had gone by after 77 seconds. Leaving aside for the moment my skepticism about Type A and Type B, how many people did he study? Were they all undergrads? Did they control for people on the autism spectrum? What was the spread (I highly doubt every single participant had exactly 58 or 77 seconds; there had to have been some overlap)? Was the study replicated? The link from the article isn't much help. Late people often have a “bizarre compulsion to defeat themselves,” wrote self-proclaimed late person and TED speaker Tim Urban in 2015. Hey, I have one of those, but I'm not a late person. Sure, I'll procrastinate stuff, but when it comes to meeting up with other people, I'm on time. Usually. He gave these poor souls a name: CLIPs, Chronically Late Insane People. You go to hell with that acronym, too. It's demeaning to people with actual mental illness. And depression often comes with low energy, making mustering the motivation to get a move on all the harder. I can relate to that, as well, but when I'm depressed, I'm more inclined to just blow something off rather than be late for it. For those left waiting, there is hope. You, too, can dictate what you’re willing to put up with. “Instead of getting angry or upset, you can take a stand and set boundaries,” she says. “Talk about what you will do if the other person isn’t on time.” For instance, tell your late friend you’ll go into the movie without them if they’re more than ten minutes late. That's way too generous. I'm not going to miss one second of the previews. You get there on time or early, or I watch the movie without you. In summary, the article might have some good points. I don't know. But perhaps it will help someone. I especially liked the part about the rest of us not putting up with other peoples' lateness. |
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