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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.
February 17, 2020 at 12:05am February 17, 2020 at 12:05am
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Since I'm finally getting around to learning a new language - however badly at this point - language articles get noticed more and thus stuck in my Blog Fodder List.
This is one of them.
https://theconversation.com/language-alters-our-experience-of-time-76761
Language alters our experience of time
Getting this bit out of the way first: no, time is not an illusion. But just like different people might estimate a mile differently (which is why we have measuring sticks), different people perceive the passage of time differently. Or the same person perceives it differently at different times.
And, to make another analogy, I've heard Europeans complain that Americans give distances in time. Like, the nearest city might be a half hour away, or a drive across the country takes three days. So at the very minimum, I'm prepared to accept that there are cultural differences in our perception of time.
My new study – which I worked on with linguist Emanuel Bylund – shows that bilinguals do indeed think about time differently, depending on the language context in which they are estimating the duration of events.
Again, no real surprise. Language gives us a particular way of looking at the world, and switching languages is like switching filters on a lens.
The logic in Aymara appears to be this: we can’t look into the future just like we can’t see behind us. The past is already known to us, we can see it just like anything else that appears in our field of vision, in front of us.
Which is the part I find really fascinating - the perception of time there is completely turned around from what we're used to. Like living in a mirror, where left is right and right is left.
A similar thing that I haven't seen too much written about is the direction of written language. Most Western scripts read from left to right, like we're doing right now. Hence we tend to order things in that direction: door number 1 is on the left, door number 2 in the middle, door number 3 on the right. To number them backwards seems at least weird, like you're about to launch a rocket via countdown.
But other languages, like Hebrew, read from right to left, so I imagine they'd order things the second way naturally.
There are interesting historical reasons for the differences in direction, and I suspect they both have to do with the predominance of right-handed people. If you're writing with ink on parchment, you want the wet ink to be in the clear, and not smudged by your hand as you go along - so you'd be inclined to pull your right hand to the right. Whereas if you're chiseling words on stone, and you're right-handed, you want to hold the hammer in your right hand and chisel in your left. Smudging isn't an issue, but angle might be. So, basically, left-handed people got the short end of the stick - sometimes literally.
None of which explains vertical scripts very well, and I don't have much knowledge about them. Anyway, to get back to what the article is saying, I think there are a lot of unconscious biases that we have, and our perception of time might very well be entangled with our perception of language.
It seems that by learning a new language, you suddenly become attuned to perceptual dimensions that you weren’t aware of before.
Again, if true (I don't know enough about any other language to have personal experience with this), that's really interesting.
But it also shows that bilinguals are more flexible thinkers and there is evidence to suggest that mentally going back and forth between different languages on a daily basis confers advantages on the ability to learn and multi-task, and even long term benefits for mental well-being.
Which relates back to one of my reasons for deciding to learn other languages at my advanced age. I mean, I almost never do anything for just one reason, and the primary reason has to do with booze (naturally), but why not make up for the brain cells lost to said booze by making my mind more flexible? |
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