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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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| It's been a while since I've ragged on the "time is an illusion" nonsense, so here we go again. Despite my issues with the wording, it's an interesting article with some things I hadn't read before. Obviously, I can't do the whole thing justice here; that's what the link is for. America's official time is kept at a government laboratory in Boulder, Colo., and according to the clock at the entrance, I was seven minutes behind schedule. Not if time is an illusion, you weren't. NIST broadcasts the time to points across the country. It's fed through computer networks and cellphone towers to our personal gadgets, which tick in perfect synchrony. For various definitions of "perfect." "A lot of us grow up being fed this idea of time as absolute," says Chanda Prescod-Weinstein, a theoretical physicist at the University of New Hampshire. But Prescod-Weinstein says the time we're experiencing is a social construct. That's misleading at best. Sure, the way we slice time up into hours, minutes, and seconds is purely arbitrary (though it does have some basis in observation), but time's going to do its thing regardless of whether there's a clock to measure it. The orbits of the planets, for instance; they make an effective clock if you know how to read it. How can I claim to know better than a theoretical physicist? It's the "theoretical" part. The smallest "things" we know of, quarks and electrons, don't experience... well, anything. But where time is concerned, subatomic particles follow laws that don't take much notice of time, if any. From that perspective, okay, time isn't fundamental, and that's the framework a theoretical physicist works in. But for everyday stuff? Time is real. The sun rises in the east (though the directions are also a social construct) and sets in the west. There is darkness, and then there is light. The consensus I've seen is that it's a bulk property of matter, related to entropy. You know what else is a bulk property? Temperature. But there aren't pseudo-mystics floating around airily proclaiming that temperature is an illusion. Any that do need to be shipped to Antarctica in a pair of shorts to see if they can wish temperature away. Real time is actually something quite different. In some of the odder corners of the Universe, space and time can stretch and slow — and sometimes even break down completely. You're going to claim time is a human construct, and then, in the exact same paragraph, use the phrase "real time?" Yes, that last quoted bit is correct, to the best of my knowledge. The thing is, though, we know exactly how time stretches, slows, and breaks down under acceleration (including acceleration due to gravity). There are equations for it. To me, an "illusion" wouldn't have that quality. Space is also different from one point to another, but only the most bearded philosophers claim space is an illusion. For many people, this unruly version of time is "radical," she says. It is, by definition and equations, ruly. Not unruly. Yes, it seems odd to us because it's outside our normal experience. But there's plenty of observational confirmation of the way time changes at different locations. By averaging a subset of the 21 clocks together, NIST has created a system that can count the time to within one quadrillionth of a second. That means the government's clock can keep time to within a second over the course of about 30 million years. At which point it'll be moot, because the Earth's rotation will have changed, and the second is based on the minute, which is based on the hour, which is based on the time it takes for the Earth to make a complete rotation during the current epoch. I expect the second will remain the same, provided we last long enough to keep measuring time. But the length of the day will gradually increase, unless something catastrophic happens. The time from this lab is used to run our lives. It says when planes take off and land, when markets open and close, when schoolchildren arrive at class. It controls computer networks, navigation tools and much, much more. And? I'm as lazy as anyone, but I still want to keep track of time if I'm doing something or meeting with someone. Governments around the world aren't just providing the time as an altruistic service to citizens, Prescod-Weinstein argues. It's about keeping society organized and efficient. It's about increasing economic productivity. Bit of a stretch, in my opinion. "Capitalism sucks, and I think a lot of people's relationship to why time is not cool, is structured by the resource pressures that we feel," she says. So she has an agenda. I'm not going to get into the "capitalism sucks" debate, except to say that, well, we've tried some other systems, and as Churchill said, it's the worst economic system, except for all the others. I do hold out some hope that we'll find a replacement, à la Star Trek, or improve it so it's not so dehumanizing as we pursue peak efficiency in the name of Holy Productivity, and pretend that infinite growth is possible. So I can relate to that agenda. But really, none of that says anything about the concept or reality of time itself. Wibbly wobbly timey wimey I can't hate an article that has a Blink reference. True time is actually much more flexible than most people realize, Prescod-Weinstein says. According to Einstein's general theory of relativity, space and time are tied together, and space-time can bend and curve. Sure, but that has no practical value for us as we slog through our daily lives. In places where gravity is very strong, time as we understand it can break down completely. At the edge of black holes, for example, the powerful gravitational pull slows time dramatically, says Prescod-Weinstein. And upon crossing the black hole's point of no return, known as its event-horizon, she says space and time flip. I've seen that finding before, and it's got lots of theory supporting it. Obviously, there's no way to experimentally verify it. Again, no practical use for us. Not yet, anyway. The Universe is expanding, and because of entropy, energy and matter are becoming more and more evenly spread out across the ever-growing void. In its final state, the Universe may end up as an inert cloud of energy and matter, where everything is evenly distributed. Yeah, I've referred to that before. They call it the heat death of the Universe, because there will be no more heat transfer, because everything is already at maximum entropy. As I noted, time is probably the result of the one-way direction of entropy. No entropy change means no time. That's if our current cosmological models are correct, which is always an active question. What this article fails to mention is that this is not an imminent existential threat. We're talking about something like 10100 years from now, which is a number so large that you don't understand it. Hell, I barely understand it, myself. As a comparison, there are fewer than 10100 atoms in the entire observable universe. That exact number is called a googol, incidentally. Not to be confused with Google, who either deliberately or accidentally misspelled it. Anyway. So time, as we understand it, has some really big problems, but it also has some really tiny ones, too. In fact, some scientists who study the microscopic interactions of fundamental particles are questioning the idea of time itself. Yes, we know, fundamental particle interactions are time-reversible. As I said up there, time is a bulk property, to the best of our knowledge. Well, I've banged on long enough. There is, as I noted, a lot more stuff in the actual article. If you have time to read it. |
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