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 |    Not for the faint of art. |
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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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| I believe in coincidence. That is, when two or more seemingly unrelated events appear to converge in a manner meaningful in some way to me or other humans, it's not because someone somehow steered the results, but because of pure coincidence. You know what would make me consider actually believing in the supernatural? If there were never any coincidences. Random chance will occasionally put two or more factors in close proximity, like when, occasionally, a cloud will cover the Sun and Moon during a total solar eclipse. (The eclipse itself is a giant cosmic coincidence, what with the Moon and Sun appearing to be about the same size in the sky.) If no coincidences happened, well, that would require a Vast Cosmic Intelligence to avoid them. Coincidences are simply the occasional ordinary workings of random numbers and/or chaos. But sometimes, I'll run across a coincidence that stretches all credulity, that is so utterly appropriate as to make me gape in speechless awe at the sheer metaphysical metaphor (or metaphorical metafizz) of it all. Well, today's coincidence is not quite on that level, but almost. For complete background, first you have to know that my link queue is, as of this morning, 48 items long, and each of the 48 items have the same chance of being selected at random. A 1 in 48 chance, to be precise. So, roughly, today's article, from the BBC, being the only one in the queue on this subject, had only about a 2% probability of being selected today. And today is the day when I (unless I fall over dead before I get to it) reach a nice round number milestone 2000-day streak on Duolingo. See the connection? It's only meaningful if one attaches significance to numbers with lots of zeros in them. But, let's be real here, most of us note such round numbers as special. But enough about that. Time to take a look at the actual article. I'm standing in line at my local bakery in Paris, apologising to an incredibly confused shopkeeper. He's just asked how many pastries I would like, and completely inadvertently, I responded in Mandarin instead of French. Ah, the age-old tradition of the humblebrag. I'm equally baffled: I'm a dominant English speaker, and haven't used Mandarin properly in years. Pretty sure that's one language I know I'll never learn. Multilinguals commonly juggle the languages they know with ease. But sometimes, accidental slip-ups can occur. I want to be clear, here: I don't consider myself multilingual. I can understand a good bit of written French. Je peux écrire des mots en français. I can't pronounce it well enough to be understood, and I can't follow spoken French well enough to understand most of it. In other words, I'm not fluent. So, any communication in French, I have to translate to English in my head, then compose a sentence in English and translate it into French. In doing so, I make mistakes, English slips in, and it becomes a kind of creole that even my New Orleans-born father would have cringed at. So I'm guessing that such mistakes become rarer as one gains fluency, but I don't know for sure. Research into how multilingual people juggle more than one language in their minds is complex and sometimes counterintuitive. I hope it's counterintuitive. That's one reason we do science: to rise above mere intuition and "common sense." "From research we know that as a bilingual or multilingual, whenever you're speaking, both languages or all the languages that you know are activated," says Mathieu Declerck, a senior research fellow at the Vrije Universiteit in Brussels. I shouldn't make assumptions about people based on their names or where they're from, but I'll point out that the official languages of Belgium include Dutch, French, and German, and English is also widely used; when I was there, I saw signs and heard speech in all four languages (sometimes a sentence would switch easily between them), and more—though I'll be the first to admit that, on hearing them, I'm not sure I could reliably tell Dutch from German. My point is only that if anyone can hold a claim to knowing about multilingualism, it would be Belgium. Or Switzerland. But I'm not visiting Switzerland; it's where my ex-wife lives. Yes, I know it's a relatively large country; shut up. "For example, when you want to say 'dog' as a French-English bilingual, not just 'dog' is activated, but also its translation equivalent, so 'chien' is also activated." Those words also come from different sources. What I don't understand, and can't be arsed to look up now, is why Spanish, linguistically related to French, uses a completely different word, 'perro'. Declerck himself is no stranger to accidentally mixing up languages. The Belgian native's impressive language repertoire includes Dutch, English, German and French. And what did I just say? Yeah, sometimes assuming doesn't make an ass out of u and ming. "The first part was in German and I'd step on a Belgian train where the second part was in French," he says. "And then when you pass Brussels, they change the language to Dutch, which is my native language. So in that span of like three hours, every time the conductor came over, I had to switch languages. Which sounds impressive, but remember, I navigated the Belgian rail system while knowing maybe four Dutch words, and one of them is "clock." Okay, "klok." Train announcements are verbal, though. The article moves on to describe some experiments that study this code-switching and its associated errors, and it's very interesting to me, but not a lot of point in quoting from it. Just one thing from the middle of that section: "The brain is malleable and adaptable," says Kristina Kasparian, a writer, translator and consultant who studied neurolinguistics at McGill University in Montreal, Canada. "When you're immersed in a second language, it does impact the way you perceive and process your native language." What I need, after 2000 days on Duolingo (a streak, I'll reiterate, even longer than my current daily blogging streak), is to find a French speaker as a coach. I mean, I've needed to do that for some time. I couldn't do it in France, because, in general, French people have no patience for that merde. Navigating such interference could perhaps be part of what makes it hard for an adult to learn a new language, especially if they've grown up monolingual. And make no mistake: it is hard. Not impossible, as some claim; that I have had any success at all disproves that. But I'm sure I'd have learned French much faster in my youth, which was wasted learning Hebrew, Latin, and computer programming (all of which I've forgotten all but the very basics of) (pun intended). Some studies have shown bilinguals perform better on executive control tasks, for example in activities when participants have to focus on counterintuitive information. I have no idea if my language learning has helped with that. Speaking multiple languages has also been linked to delayed onset of dementia symptoms. I don't wish to die, but it would be preferable to dementia. And of course, multilingualism brings many obvious benefits beyond the brain, not least the social benefit of being able to speak to many people. These days, smartphones can assist with translation. I witnessed people using them in Europe. They're nowhere near perfect, but I'm sure they do in a pinch. Not only can you get verbal translations, but also text translations. All great technology, but no substitute for learning, in my opinion. And so, I continue to learn. |
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