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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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Clearly, it's sorcery. Or maybe ghosts.
Actually, what "no one can explain" is why I fell for the clickbait headline. Maybe because I expected better from Scientific American.
In December 2003, to commemorate the 100th anniversary of the first flight of the Wright brothers, the New York Times ran a story entitled “Staying Aloft; What Does Keep Them Up There?”
One could say it doesn't much matter, because it's obvious that they do stay up. The vast majority of the time, anyway.
But that's not science. This is like the old nonsense about how bumblebees can't fly (spoiler: they actually can).
To answer it, the Times turned to John D. Anderson, Jr., curator of aerodynamics at the National Air and Space Museum and author of several textbooks in the field.
Okay, I can accept those credentials, for now.
What Anderson said, however, is that there is actually no agreement on what generates the aerodynamic force known as lift. “There is no simple one-liner answer to this,” he told the Times.
"There is no agreement" is not the same thing as "there is no simple answer."
In the interest of full disclosure, I had, for a long time, been under the impression that it's all about Bernoulli's Principle (airfoil causes air above the wing to have higher velocity and thus lower pressure), though it never really seemed to explain everything. And it doesn't. For instance, you know those old planes that could fly upside down? Can't be explained by airfoils alone.
By far the most popular explanation of lift is Bernoulli’s theorem, a principle identified by Swiss mathematician Daniel Bernoulli in his 1738 treatise, Hydrodynamica.
Told you so.
The article goes into this in more depth, of course, but I won't reiterate much of it here. Basically, there are other factors involved as well.
The controversy, as far as it goes, seems to be about how much each of those different factors contribute. There's also unstable, chaotic effects at play, which are poorly understood right now.
But that's not why I linked the article. I mean, sure, this stuff's fascinating to me, but I don't expect everyone to be riveted by the article or whatever. No, my main points here are to show that a) just because you don't fully understand something doesn't mean it's not real; and b) I'm really annoyed that SciAm would use a headline like that. It fosters distrust in science, something that's pervasive enough in the world right now. "Why should I trust scientists?" I can hear the willfully ignorant saying while clutching a Coors Light. "They can't even explain how airplanes fly!"
It's true that science sometimes gets things wrong. I'm not contesting that.
It is even more true that without it, we'd get a whole lot more wrong. |
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