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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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Today, another bit about our solar system, this one certain to be controversial.
The article, from The Atlantic, is two years old, but Pluto's label hasn't changed since then.
In 2006, astronomers gathered in Prague to consider a very basic question: How many planets are in our solar system?
They needed to Czech in with each other.
For many Americans, the names of planets were some of the earliest scientific facts we learned, and that there were nine of them seemed like a basic truth of existence.
This is a failure of science education in America.
There was no consensus among [astronomers] then, and there is none now. Ask one astronomer and they’ll sigh, before saying that it’s time to stop dredging up the past and move on. Ask another and they’ll say the matter desperately needs a do-over. They agree on so much about the cosmos, but on a matter that seems as though it should be straightforward, some of them might as well exist in two different solar systems.
You know, I bet you can get them to agree, within a significant figure or so, about the mass of Pluto, the length of its year, the period of its mutual orbit with Charon, and so on. This is, as I've noted before, a classification problem, akin to deciding whether to call a stone a "rock" or a "pebble."
In 1801, astronomers deemed Ceres, a rocky object they had spotted between Mars and Jupiter, a planet, but 50 years later, after further observations, they designated it as an asteroid.
This is the bit no one remembers. Unsurprising, as I don't think anyone now was around then.
Then, in the 1990s, astronomers started finding icy objects beyond Neptune. “People who were paying attention immediately said, Oh, we get it. This is what Pluto is,” Mike Brown, a Caltech astronomer who discovered one of these celestial bodies, told me. “Pluto is not this oddball at the edge of the solar system; Pluto is part of this larger population.” In 2000, when the American Museum of Natural History, in New York, opened a new space wing, curators tagged Pluto as part of the Kuiper belt, the disc of objects floating past Neptune, a decision that prompted a flurry of hate mail from Pluto fans.
Make Pluto Planet Again!
To Brown, the International Astronomical Union’s judgment was brave, considering how the public would perceive the loss of a beloved world.
Pluto isn't lost. We've even sent a space robot flying by it. It hasn't gone anywhere. Well, I mean, sure, it's progressed a bit along its orbit, but it's still in the orbit.
Some planetary scientists, whose work doesn’t center on such orbital details, say that there’s much more to a planet than that. They argue that Pluto has a host of characteristics that qualify it for planethood: an atmosphere, geological activity, even five of its own moons.
"A hot dog is meat between hunks of bread. Therefore, it is a sandwich, has always been a sandwich, and will always be a sandwich!" "No, it's a taco." "Heretic!"
"You call these things 'mountains'? Come out west, I'll show you mountains!"
"That's a river." "No, it's a stream." "No, it's a creek."
New Horizons flyby in 2015 make discounting Pluto difficult. The mission gave humankind its first close-up of Pluto, uncovering stunning topography, such as a massive heart-shaped glacier made of nitrogen. Fran Bagenal, a planetary scientist at the University of Colorado at Boulder, emailed me a picture from the flyby and wrote, “Doesn’t this look like a planet to you?”
This is Ceres.  It orbits the sun. It looks like a planet to me, too, being round and having surface features and shit, although admittedly not as visually striking as those of Pluto. As noted above, Ceres was a planet for half a century. As with Pluto, changing what it was called didn't change Ceres.
For all their disagreements, everyone I spoke with was on the same page about one thing: that the question of planethood has no bearing on whether Pluto is a fascinating place to study.
It could be argued that this "controversy" worked very well to get more people interested in space, which I absolutely approve of.
For anyone still wondering why it matters what we call a planet, dwarf planet, asteroid, etc., well, it really doesn't. But it's nice to have a working definition as we discover more and more bodies orbiting other stars. While, to the best of my knowledge, we don't have the technology to detect extrasolar dwarf planets, these questions of classification will go with us to the stars... assuming we make it that far. |
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