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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.
January 31, 2020 at 12:07am January 31, 2020 at 12:07am
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Did anyone get here by clicking through from "Born Funny"  by Annette in France  ? If so -- Hi! I'm Waltz; since you have the perspicacity to be reading the Comedy newsletters in the first place, you probably already know me. If not, short version: I like the indoors, live an alcohol-positive lifestyle, love science and philosophy and music and movies, and am owned by cats. Also, all the stuff at the top of the blog is meant to scare off math-phobes; "Complex Numbers" is a multi-level play on words and I rarely actually talk about mathematics in here.
Thanks for linking this in the NL, Annette in France . Now I'm under pressure to be funny, so I probably won't be.
Since the 30DBC ended yesterday, and I now have an even bigger backlog of blog fodder, let's go straight to the stars.
https://www.theatlantic.com/science/archive/2020/01/betelgeuse-supernova/605251/...
The Biggest Celestial Event of the Year Could Happen Tomorrow
... or, well, maybe not for 100,000 years
Spoiler alert: It didn't happen tomorrow. Article was dated January 23, and I'm sure I looked up between then and now, and still nothing. But since that involves going outside, I can't say it didn't just happen. You'd think that, when it did, it'd be all over the internet, but no, everyone's too busy freaking about about Kung Flu.
What is "it?" A supernova.
Sometime this week, you might walk outside in broad daylight, look up at the sky, and see a luminous orb as bright as a full moon. Only it wouldn’t be the moon. It would be something far more explosive: the dazzling aftermath of a cataclysm hundreds of light-years away.
Now I'm depressed that it probably won't happen in my lifetime. Because that sounds cool as shit.
In the night sky, the constellation Orion is most well-known for his belt, a row of three luminous stars.
Alnitak, Alnilam, and Mintaka. No, I didn't just look that up. Orion is probably the best-known and most easily identified constellation of all - the Big Dipper being only part of a constellation - so at one point I knew the names of all its visible stars. Now it's just those three, plus Rigel and Betelgeuse. I could look up the rest again, but can't be arsed.
Those names, like many star names, are of Arabic origin. While Europe was languishing in the dark ages of superstition and deliberate ignorance, the Arab world was busy giving names to stars.
For the last few months, though, astronomers around the world have been particularly interested in his right shoulder, the home of a star called Betelgeuse, one of the brightest stars in the sky.
Just to be clear, we imagine Orion as an anthropomorphic figure facing us, so his right shoulder is on our left as we look up at it whilst facing south-ish.
The supernova wouldn’t harm Earth. Betelgeuse isn’t the sort of star whose demise would produce radiation that could roil the planet’s atmosphere. At about 650 light-years from here, Betelgeuse is nearby on a cosmic scale, but thankfully not close enough to cause any damage.
I just wanted to include this paragraph in case you can't be arsed to click on the link.
So how might people react? Judging by what happened in New York about a year ago, there would be confusion, even panic.
[insert Hitchhiker's Guide to the Galaxy reference here]
And sigh. Because if people weren't so damn willfully and boastfully ignorant about these things, they wouldn't panic.
A similar scenario would likely play out online in the case of a surprise supernova, with NASA and other science institutions leading the awareness campaign.
Right, because when there's a big scary light in the sky and you haven't been reading my blog (or any of numerous science articles available on the web), you're going to trust what a government agency says. "They're covering it up again! Aliens! Armageddon! Zombies!"
The star might explode tomorrow or in 100,000 years, says Stella Kafka, the executive director of the American Association of Variable Star Observers and an astronomer...
And I'm just including this bit because if I were writing a novel featuring an astronomer and executive director of [insert astronomy organization here], and I named the main character Stella Kafka, my editor would insist that it was unrealistic and I should change it to something less godsdamned perfect.
On top of that, astronomers don’t have any proof that the mysterious dimming is a precursor to a supernova.
In journalism, they call this "burying the lede."
When stars explode, they release a cascade of newly forged elements into space. These elements glide across the universe inside particles of dust, settling on whatever they encounter. Astronomers have detected this stardust all over Earth, inside mud on the ocean floor and snow in Antarctica. It is these explosions and the cosmic droplets they unleashed that helped give rise, over eons, to other stars, planets, and, in our case, life. Someday, a bunch of stardust might look up at the sky and see it happening all over again.
That's very poetic and a great way to end an article like this one. It's not a new interpretation, and some might even call it trite because it's been repeated, in one form or another, for decades now. But I never get tired of its philosophical power
To add a bit more detail: when the Universe was a baby, it was roughly 3/4 hydrogen and 1/4 helium by mass (gross oversimplification, but the point is, very few heavier elements existed then). This gas clumped into galaxies and stars, and within the stars, fusion began to create slightly heavier elements like carbon and oxygen. These stars exploded, spewing their contents into the void, contents which then formed other stars - still mostly hydrogen, but with some heavies (what astronomers call "metals"). Those stars, in turn, exploded, etc.
The thing about fusion is that knocking nuclei together to create everything up to iron in terms of atomic number releases energy, while iron and above absorbs energy. There is a lot of energy in a supernova, so you end up with a lot of iron and a few other heavy elements like, e.g., gold or uranium. So the supernova blasts these elements into interstellar space, and they later accrete around a nascent star, and some of them form planets, and some of these planets (at least one, anyway) go on to develop complex life that utterly depends on a wide range of elements, including iron.
There's probably no helium in you, unless you just inhaled some so you can talk in a squeaky voice, so every single atom in your body (except the hydrogen) was forged in the heart of a star and driven across the galaxy by the incredible forces of a supernova. Hell, even if you have been sucking on a balloon, that helium is probably the decay product of some radioactive element that was star-born.
I don't care how many times I hear it. That's fucking cool.
One final thing: don't get confused by anything that goes something like "Betelgeuse might have already exploded, but the light hasn't had time to reach us yet!" This is technically true, but irrelevant. It's not like we have a warp drive to go check out the vicinity and see if it's "already" exploded (and if we did, the Enterprise could be in for a really rough ride when it drops out of warp). For any reasonable linguistic value of the present tense, the supernova happens when we see it, and not before.
I wasn't going to continue with the musical tie-ins, but this one was just too obvious.
Well maybe it is just the time of year
Or maybe it's the time of man
I don't know who l am
But you know life is for learning
We are stardust
We are golden
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