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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.
February 28, 2008 at 4:58pm February 28, 2008 at 4:58pm
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My February blog calendar has a nice checkerboard thing going and I'm loath to mess that up, right now. I've hit every even-numbered day since the 4th, which is the reason for the neat little pattern.
Leap years mean three things to me: Election year (about which enough has been said in here, for now), the Olympics (about which I don't care) and that February and March don't have the same date/day mapping. Had I been born on a leap day, it would probably have some sort of significance for me.
Calendars in general, though, are fascinating in many ways. First is the mathematical gyrations necessary to keep the calendar pretty close to the Earth's orbit. These calculations are even more interesting for lunisolar calendars, which often have leap months instead of days - which is why Jewish holidays meander about on the Gregorian calendar; they use a lunisolar calendar, ensuring that I always have to work to figure out when to wish my aunt a happy (whatever)day.
The Gregorian calendar itself was a modification of the Julian calendar, so called because it was instituted by Julius Caesar. Now, Caesar was arguably the most brilliant general in history, and the second-most brilliant politician (the top spot going to his successor), but as an astronomer he likely didn't know his tropics from his sidereals. Still, not only did he manage to get his name attached to a calendrical system, but also to one of the damned months (damned, that is, from a Northern Hemisphere perspective, as in "Damn, it's a hot and humid July).
Incidentally, lots of things have been named after Gaius Julius Caesar, but the procedure known as "caesarian section" was not. No, Caesar was named Caesar (probably) because he was born using this procedure (about 2000 years before epidurals were invented). And given that since his time, there was an emperor titled after Caesar right up until "Anastasia screamed in vain," that's about 2000 years of legacy for one breach birth... but I digress.
So the Julian calendar, in turn, was a modification of the earlier Roman calendar, which had pretty much the same month names as we have now (with the obvious exception of July and August). The Roman calendar had what they called an intercalary month, which sounds more like a tooth decay prevention device than a system for aligning lunar cycles with solar, instead of a leap day.
As another aside, the birthday of Augustus Caesar, to whom I alluded to as probably the greatest politician who ever lived, is September (not August) 23. That is also the birthday of the greatest musician who ever lived, Bruce Springsteen. I'm just saying.
The original Roman Calendar, which would be reformed first into Julian and then into Gregorian, began in March, not January, to align with the spring equinox - which is why the tenth month of our calendar (October) has a prefix more usually associated with eight than ten, further confusing any aliens that might be trying to figure us out prior to invading. It most likely had its own origin in more ancient lunar calendars, which is why we still call groupings of days "months," after the moon.
Which begs the question: why the shift from lunar to solar? We can see the phases of the moon, even today, a whole lot better than we can see the Earth's position relative to the sun and stars. Simple: herders and hunters need to know moon phases so they can know when they might be able to see at night, but planters need to know the seasons (ruled by the sun, not the moon) so they can know when to plant and harvest. The shift from the former to the latter resulted in the modification of lunar calendars to solar.
At least, in Europe. Elsewhere, lunar calendars are still in use, notably the Jewish, Islamic and Chinese calendars. And while we (of European descent) were still trying to figure out this whole "earth revolves around the sun and not the other way around" thing, the Mayans were working out a solar calendar more accurate than ours got to be until we invented atomic clocks.
And that's pretty damn cool for a culture that couldn't quite figure out what that "wheel" thing is good for. |
February 26, 2008 at 7:13pm February 26, 2008 at 7:13pm
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This is not going to turn into a political blog, I promise. But I found this article refreshing, all things considered:
http://thecaucus.blogs.nytimes.com/2008/02/26/mccain-repudiates-hussein-obama-re...
A conservative radio talk show host who helped introduce Senator John McCain before a rally here Tuesday used Senator Barack Obama’s middle name, Hussein, three times, while disparaging him, prompting Mr. McCain to apologize and repudiate the comments afterward.
...
“It’s my understanding that before I came in here a person who was on the program before I spoke made some disparaging remarks about my two colleagues in the Senate, Senator Obama and Senator Clinton,” he said. “I have repeatedly stated my respect for Senator Obama and Senator Clinton, that I will treat them with respect. I will call them ‘Senator.’ We will have a respectful debate, as I have said on hundreds of occasions. I regret any comments that may have been made about these two individuals who are honorable Americans.”
I don't know if that's just more political maneuvering, but I like it. I like it a whole lot better than "My worthy opponent screwed a dead mule in the town square, and I have the pictures to prove it." |
February 24, 2008 at 9:13pm February 24, 2008 at 9:13pm
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I really, really hate election years.
Like assholes, everyone has an opinion, and most of them stink. I'd care more if I thought caring would make a difference, but it doesn't. Voting can make a difference, but only if enough people do it.
Lately I've been hearing scandals about all the front-runners, together with really good reasons why none of them should be allowed in office.
You know what? Dig deep enough, you'll find dirt on anybody. Anybody. And it's always qualified, like "Your candidate had sex with a dead mule in the town square. I mean, I could handle it if it'd been a LIVE mule - but a DEAD mule? That's horrible!"
Don't expect candidates to be perfect, or even internally consistent. You're not - why should they be? |
February 22, 2008 at 9:38pm February 22, 2008 at 9:38pm
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My wife and I had planned to go to SheVaCon, the nearest science fiction / fantasy / gaming convention to us, this weekend. Of course, my back, while improving, still gives me periods of excruciating agony. It's like someone's pouring molten lead down my leg, which is a step up from the molten iron of weeks past.
And then while we're getting ready, she falls down the damn stairs and twists her ankle.
Oh, we went anyway - no sense wasting the hotel and con fees - but neither of us can really lift anything, and we're both moving like 90 year olds. Actually, I've seen a lot of 90 year olds who could outpace us right now.
So we're here, but we can't really do anything. I hope that later I'll have a chance to mingle among the hordes of geekdom here, avoiding LARPers and catgirls while coming up with ideas for my next Fantasy newsletter and looking for one of the many writing workshops...
...speaking of which, it seems next year's writer guest of honor will be Larry Niven.
I don't care if I'm in a fucking full-body cast next year; I'm totally going to THAT. |
February 20, 2008 at 10:34pm February 20, 2008 at 10:34pm
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http://antwrp.gsfc.nasa.gov/apod/ap080220.html
I was concerned, today, that the cloud cover would preclude my viewing of the total eclipse of the moon this evening. As with much in life, I hoped for the best but prepared for the worst... but it's a wonderfully clear night, and the moon around 8:30 looked like it had a big chunk bitten out of it.
It's at the times of eclipse that I can best visualize how the solar system is put together. In this case, we're treated to a view of Saturn just a few moon-widths away (the moon subtends an arc of 0.5 degrees of sky; thus, two moon-widths is about one degree of arc. There are, of course, about 180 degrees of arc across the visible celestial equator, and a total of 360 degrees, corresponding to 360 degrees of longitude. Why 360? Largely because there are 365 days in a year, and 360 is the closest number with many factors.
They came up with that system in ancient Babylon, or perhaps inherited it from an even older culture. The ancients were pretty good with numbers, but fractions sometimes stumped them; hence, they looked for numbers that were easily divided. That's why there's 24 hours in a day instead of some multiple of 10; it's divisible by 1, 2, 3, 4, 6, 8, 12 and, of course, itself. 24 itself goes neatly into 360. And then there's the mystery of why there are seven days in a "week" and not six or eight; I think that calendar evolved separately from the solar calendar, as weeks of seven days work relatively well in marking off lunar cycles. Perhaps, long ago, weeks alternated between seven and eight days, before some bright caveperson noticed: seven days in a week; seven moving celestial bodies. Coincidence? He or she didn't think so. And so the days of the week took on the attributes of the Sun, Moon, Mars, Mercury, Jupiter, Venus and Saturn (in that order), disengaging the week from its relationship to the moon. Just a thought I had; I don't know if there's any reality to it.
On a more personal note, things are starting to turn around a bit, for me. My back continues to improve. We're starting to get some jobs dribbling in at the office. My dad's going to be released from the hospital soon, perhaps even tomorrow, to go back to his assisted-living place - though he'll require more nursing care.
Well, time to go check on the moon again - it should be at its darkest right about now. It can only get brighter from here. |
February 20, 2008 at 7:03pm February 20, 2008 at 7:03pm February 18, 2008 at 4:46pm February 18, 2008 at 4:46pm
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Officially, today is George Washington's Birthday observance. Calling it Presidents' Day (or Presidents Day or President's Day) is a commercialization, promoted mostly by retailers so people will shop there on their day off.
I talk about Jefferson more, here, because I live in a town indelibly stamped with his seal. But I spent my childhood much closer to the home of George Washington.
I never much liked George Washington. I remember from an early age thinking he was a pompous ass. I mean, come on: the cherry tree incident? Chucking a dollar across the Rappahannock?
I derive great pleasure from imagining Jefferson upending a stein of ale onto George's powdered wig.
And look, the guy was a traitor, you know. Had the Revolutionary War gone the other way, that other George would have had him pilloried, drawn and quartered. Oh, I guess he got "quartered" anyway.
Where Jefferson was witty and brief, Washington was sober and verbose. Where Jefferson was a populist, Washington was a federalist. Where Jefferson was a diplomatist, Washington was a militarist. And don't forget: from all reports, Washington was a lousy general, winning more out of luck than skill. I mean, first he FOUGHT the French (and lost), then he fought WITH the French against the British - and still kept losing. Out of nine battles he fought, he won three. THREE.
Of course, one of them allowed us to drop that British accent and spell color without a 'u.'
But I can't deny he was a great President - partly because he defined the office for everyone else, but partly because pompous assery works pretty good in high office.
Did you know my town celebrates Jefferson's birthday? Yep, they get the day off every April, in addition to Washington's Birthday.
Well, Washington's Birthday, as observed, rarely coincides with mine, but when it does, I really, really hate being overshadowed by a pompous ass whose main visible legacy is a slave plantation, a stone face out in the middle of nowhere, and a 555'5-1/8" phallus.
Wait, it says here he grew hemp on one of his plantations. Nevermind - ol' George was cool.
Anyway, I don't get a 555'5-1/8" phallus, but I got the next best thing, this year - a birthday forum!
You guys rock. Thanks!  |
February 16, 2008 at 6:51pm February 16, 2008 at 6:51pm
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I promised some people an update, but it's going to be a quick one.
Originally we thought maybe my dad would get out of the hospital and head back to the assisted living place he's been at for about five years now. But it turns out he doesn't have a simple UTI; it's complicated by a problem with his bladder - something to do with diverticulosis, which I didn't think affected bladders, but that's what the doctor said showed up on the CAT scan.
Apparently, the best treatment for this is major surgery - IF you're young and healthy. Failing that, there is no good treatment for it. They're going to try some things, but the doctor emphasized that anything they do is, well, kind of experimental, like when you fuck around with your computer until you figure out why it keeps crashing.
So I don't know what's going to happen. The doctors don't know what's going to happen. And I don't really know what to hope for, so instead of hoping, I consider the possible outcomes and make plans accordingly.
I need to ask you all a couple of favors, though. First, while I appreciate the thought, please don't pray for my dad - none of you know him, and no matter what happens, nothing good can come of this. Second, soon - possibly tomorrow - I'm going to drop my cloaking device and be accessible via IM again. When you see me, please don't inundate me with questions about this, my own health, my business, etc. If I'm online, that means I don't want to think about it for a little while. I appreciate the concern and empathy I've received in recent comments, and would like to repay it by just being myself (to the extent that's possible online). If you're genuinely interested (or even if you're not), I will post updates in my blog when I feel like it.
But I'm also behind on my weird-ass news and funnies links, so I hope to catch up on those too  |
February 14, 2008 at 9:28am February 14, 2008 at 9:28am February 12, 2008 at 6:47pm February 12, 2008 at 6:47pm
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My dad's not getting any better - though he doesn't seem to be getting any worse; and today, the doctor - if you can call a 28 year old resident punk (think Dr. Chase from "House," only taller, darker and handsomer, so I have to keep him away from my wife) a doctor - was going over my options, rights and responsibilities as the person who has to make all his health care decisions.
I don't think it's going to come to that this time around, but I need to know these things just in case - and so I can make informed decisions when the time does come.
I don't talk about this stuff, much. Especially not online. I do this online thing because of the anonymity of it all, but there's not much anonymity here. People know me, or think they do. It's hard to talk about some of this stuff - not because of the people who know me, but because of the ones who don't, or think they do. So I withdraw behind session privacy and IM blocks, for now. It's nothing personal. Nothing personal, that is, for any of you, but very personal for me.
Dad was like that - kept personal stuff private. He liked to play his cards close to his chest. Actually, he didn't like to play cards at all, though he taught me the basics of poker along with how to catch a fly ball - which, when I look back on it now, is very odd, since he never showed any interest in sports whatsoever - and bait a fish hook. You know, those things that fathers are rather obligated to do when they have sons, and it might be a better world if more of them did.
No, Dad's game of choice was chess. Chess has boundary conditions and rules. There's no element of luck, unless you consider "hoping the other guy doesn't notice where my rook is" an aspect of luck. The rules are fixed, set, and not subject to modifications or interpretation. He taught me chess at a very young age, and I don't think he ever "let" me win, but it got to where we'd play just about every night and I got better and better until most of the time I could win, or at least force a stalemate. I remember once after a long winning streak on my part, he checkmated me and I was just young enough to get mad about it. I walked away and punched the wall, leaving a dent that's there to this day.
But I was just old enough that the next night, I was back across the table from him. I don't recall who won that particular game, but I do know I was never a sore loser again after that. It wasn't that he punished me for the dent in the wall; no, it was his disappointment that I couldn't take losing like a man. I think I was ten or eleven at the time, maybe.
Life, however, isn't a chess game. It's not a poker game, either; it's not a zero-sum game, or even a game at all, as much as I'd find it amusing if someone put "GAME OVER" on my future tombstone. No, life is more like the picture at the top of my blog - fractal, generally unpredictable, chaotic. We know about taxes and death, but not how much or when, and everything else is pretty much optional.
See, a little while ago, I was talking about my dad in the past tense. But that's because the father I knew left long ago. He left when the Alzheimer's started to kick in. I know they all say that even those with Alzheimer's respond emotionally, but what good is that? Dad wasn't the emotional type. He was a sea captain, and a chemist, and an investor in the stock market; an intellectual whose response to the chaos of the sea or the fluctuations of the stock market was to stay the course, avoid storms and play it safe. Emotions were a luxury for men of his generation. If he can't engage in rational, intellectual discourse, he's not himself.
To me, Alzheimer's is the ultimate argument against the existence of a god. Where other people might turn to their faith to get them through situations like this, I ask, "what kind of god does this to people?" And don't give me shit about being tested, or never being given something we can't handle, or life as miracle crap; it won't wash, not after I saw my mother suffer through years of dementia, followed immediately by my father's decline. Still, however, and nevertheless, I found myself wishing I had a rabbi to talk to about all this. When I examined why, the answer came back in several parts: One, to satisfy myself that my father's religious beliefs (such as they were) are being accommodated; two, because I've always thought Jews have the healthiest relationship with death; three, because I feel like talking to someone about it, and all the blogs and shrinks in the world don't quite do it for me.
Well, it doesn't really matter, not right now. Chances are, he'll pull through this bout of illness on the courses of antibiotics prescribed by his doctors (it turns out he doesn't have the flu, probably), and go back to living at the Alzheimer's care facility where he's spent the last five years. But he wasn't diagnosed five years ago; no, it was closer to nine years ago that the diagnosis was made, and the average lifespan after an Alzheimer's diagnosis is seven years. Nine is remarkable. 14 would be amazing. 16 would be a miracle, if I believed in miracles - let's just say "statistically unlikely in the extreme." So, sometime in the next seven years. Sometime before I'm 49.
Dad was 49 when I was born.
So the board has become sparsely populated; many of the pieces have been captured. I'm not sure how many moves before checkmate, but its inevitability hangs over the playing field.
Aw, fuck the damn metaphor. The whole point is that I have to face facts, and steer a course through the rough waters to-
Okay, enough with the metaphors. In addition, my back isn't improving very quickly, and I'm still in pain nearly all the time. Plus, I've got business worries (thanks to the housing slump, since housing drives my industry), and I have my father's finances to deal with in addition to his health care - and that demands much of my attention right now, so I can try to keep his land intact instead of selling it or developing it to pay the damn taxes.
Still found time to vote in the Virginia primary today, though I still dislike all the candidates. But my dad would have, so can I do less? |
February 10, 2008 at 7:26pm February 10, 2008 at 7:26pm
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The wind has been crazy here, today. It started sometime during the night, whistling between the houses and threatening to spin my attic vents right off.
When I woke up, at around the crack of noon, it was still going on. I figured I'd grab some breakfastlunch and go write this week's Comedy newsletter.
That, of course, is when the power went out.
Now, I don't have my truck this weekend. A friend of mine is using it to move. My wife was out in her car, so I was stuck. I worked some crosswords for a while, but then started feeling antsy about getting the newsletter in. So I called my wife to drive me to my office. I figured my office is on a buried line, with less chance of getting knocked out.
Indeed, the lights shone bright as I entered the office. I booted up my computer, checked some email, started looking for items to include in the newsletter (funny things about romance, mostly - it's the Singles Awareness Day issue of the Comedy newsletter), and the power went out.
Fuck.
I called my wife, and eventually she showed up to pick me up. "Power's back on at home," she says. It figures; I could have waited. Slept some more, or something. Had a cigar. Whatever.
Before we went home, though, we visited the hospital, where my dad's still being treated for his UTI. In the room, golf is playing on the TV, so of course my dad's sound asleep. I asked the nurse if there had been any changes or anything; she paged the doctor.
When the doctor called back, she said, "I don't know too much about that patient. UTI. Tested positive for influenza. Going to have to keep him at least a couple more days. Did they make you wear a mask in the room?"
What? "No. This is the first I've heard about the flu. I just walked in the room. There wasn't any sign about masks or anything." Besides, some grossly obese guy was wheezing in the room's other bed.
"Oh, and there's some question about whether we can send him back to assisited living or if he'll have to be moved to a nursing facility."
Crap. Well, things are what they are, and I'll have a chance to verify all this with his assigned doctor tomorrow.
But I got home and did the Comedy newsletter anyway. Sometimes you just have to be funny, even if there's nothing left to laugh at. |
February 8, 2008 at 5:26pm February 8, 2008 at 5:26pm
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Looks like I'm scheduled to do next week's Comedy newsletter. I was at a loss as to what to do, since there's not much room for comedy in my life right now, until I realized: it's coming out right around Singles Awareness day, and if that's not a theme ripe for exploitation, I don't know what is.
They say when you're at low points in your life, that's when you need humor the most. I guess I'm not at a real low point yet, but I have been twisted every which way this week, between finances and work and my dad being sent to the emergency room. He's pretty much stabilized now, but when a 90 year old man gets sent to the ER, it's rarely a laughing matter.
It made me realize that there's a few things I need to take care of, like making sure I know all his life insurance documents, military service information and so on. There are also issues related to his estate. And of course, thinking about all this has gotten me thinking about my own impending doom (which will probably be after his, but there are no guarantees except taxes and... well, you know) and how unprepared I am for it. I mean, no one is ever prepared for it, but I haven't even made out a will or advance medical directive or anything. Hell, I haven't even decided whether they should play Dark Side of the Moon or Darkness on the Edge of Town at my funeral yet. Maybe Pink Floyd for the funeral and Springsteen for the wake. Yeah, that'll do.
That, and I have a birthday coming up too, just a few days after Singles Awareness Day. One year for V-Day, my wife got some roses for me, painted them black, and had them delivered to my office. Now, THAT was funny. No colors anymore; I want them to turn black. This year, she wants to have our single friends over for Singles Awareness Day, but I don't think she's put as much effort into planning my massive, gala birthday party (at which I want them to play Dark Side of the Moon and Darkness on the Edge of Town and maybe some Zeppelin to cap it off) as she has into planning V-Day.
My birthdays have been crappy, anyway, since I turned 7 and my grandmother died the next day. She's buried in New York, where my mother joined her about a quarter-century later. They're in the same graveyard as Andy Kaufman (if, that is, Andy Kaufman was actually buried.) My dad doesn't want to be interred there, though. He wants to be buried at Arlington, alongside a million other veterans. I guess he wasn't much of an Andy Kaufman fan.
Well, I've managed to cover birth, life, love and death all in one short blog entry, and I've even given myself ideas for next week's Comedy newsletter. Guess it's not such a bad week, after all.
"There is no dark side of the moon really. Matter of fact it's all dark." |
February 6, 2008 at 5:41pm February 6, 2008 at 5:41pm February 4, 2008 at 5:36pm February 4, 2008 at 5:36pm
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Much discussion today has been given over to the Giants' victory over the Patriots yesterday.
I'd hardly even know about it if I hadn't made the mistake of hitting the grocery store around 6 pm, and had the honor of standing in line behind swaying men and sweaty women with carts full of beer and chee-tos.
I know some people claimed not to care about the game, and only watch it for the commercials. Me, I say: Commercials are the reason I don't watch television, so there's yet another reason not to watch the superbowl.
The other thing about the superbowl is that it started the year I was born, so I always know exactly how old I'm going to be mere days after the event - as if I need another reminder. Hell, Star Trek started that year, too, and it's not around much anymore.
What gets me is just how much this matters, and I just don't understand why. I used to watch football, back in the late 70s / early 80s. Like listening to country music, it was a phase I went through. I stopped around the time the Giants broke Redskins QB Joe Theisman's leg in several places. I'm sure a lot of people watch just hoping for another spectacular injury like that one. And I admit, having the underdog team come from behind at the last moment can be pretty exciting - I saw it happen to my college football team a couple of times (and saw them get hosed by it even more).
But I got to where I Just Didn't Care. How, I wondered, did UVA having a winning (or losing) team have anything to do with its worth as a school? Why should I do anything to support the Redskins?
And they report this crap like it's news, like it matters. Same as with Britney Spears' latest shenanigans (I still say the government has her on the payroll as Official Distraction). It's entertainment. It's not news. It's circuses, and chee-tos are the 'bread.'
But I'm open-minded about most things. I'm willing to admit that maybe I'm wrong. Maybe there's something I'm missing; after all, it seems like every other American was glued to the TV screen yesterday. Maybe it should matter to me. So tell me, honestly: Why should I care? |
February 1, 2008 at 12:58pm February 1, 2008 at 12:58pm
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Aren't we all tired of the American Presidential race by now? I know I am, and I don't even watch television, which I understand is the worst offender. I mean, of course it's important to make such a difficult choice armed with all the latest important information, such as who does Hillary's underarm hair and how much Obama's bowel movements weigh, but there might be a such thing as... too much information?
Well, I finally found a candidate I can support, someone who removes the need to vote for the lesser of two evils - and renders such questions as "Does John McCain color his hair?" moot:
http://www.zod2008.com/ |
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I'm more than happy with Carrion Luggage, however.
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