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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.
September 28, 2023 at 10:04am September 28, 2023 at 10:04am
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England invented English. Here in the US, we perfected it.
Mostly, I'm just sharing this because I find it intriguing. Though, as usual, I didn't do extensive fact-checking.
English settlers faced with unfamiliar landscapes and previously unknown plants and animals in the Americas had to find terms to name and describe them.
Gosh... if only there had been people here who had already named such things.
They sometimes borrowed words from Native American languages.
"Sometimes."
By the time of the American Revolution, English had been evolving separately in England and America for nearly two hundred years, and the trickle of new words had become a flood.
Given all that, it's truly a wonder that we're mutually intelligible. Most of the time.
Corn offers an example of how English words evolved in America. Before 1492, the plant that Americans call corn (Zea mays) was unknown in England. The word corn was a general term for grain, usually referring to whichever cereal crop was most abundant in the region. For instance, corn meant wheat in England, but usually referred to oats in Ireland. When American corn came to Britain, it was named maize, the English version of mahiz, an Indigenous Arawakan word adopted by the Spanish. When the first colonists encountered it in North America, however, they almost always referred to it as corn or Indian corn, probably because it was the main cereal crop of the area.
Interestingly, the Arawak weren't North American, and apparently their staple food crop was cassava root, not maize. Yeah, it gets complicated. Also, the French word for it is maïs, pronounced similarly to maize. As with aubergine and courgette, this is one of those food words that sometimes trip up US/UK relations because the British names are closer to the French than they usually like to admit.
Much of the landscape of North America was new to the English, so many early word inventions applied to the natural world. Often these simply combined a noun with an adjective: backcountry, backwoods (and backwoodsman), back settlement, pine barrens, canebrake, salt lick, foothill, underbrush, bottomland, cold snap. Plants and animals were similarly named, for instance, fox grape, live oak, bluegrass, timothy grass, bullfrog, catfish, copperhead, lightning bug, garter snake, and katydid (a grasshopper named for the sound it makes).
Now, see, that last one is another example of me having been wrong about something, and I can admit it. We have katydids here; I see them every summer. The cats like to try to catch them, which I discourage, because katydids are cool. But I'd never really heard whatever sound they make, so I assumed (yes, I know the line about assuming, but shut up) that it was a corruption of "caryatid."
Americans repurposed other English words as well. For example, bug, which meant a bedbug in England, broadened to cover any insect...
Here's where I get pedantic: in science, "bug" refers to a subset of insects, ones with specific mouth morphology. Here in the American South, "bug" can refer to pretty much anything with an exoskeleton that lives on land, including spiders, which of course aren't insects. Other usages restrict "bug" to pests, such as midges, mosquitoes, or cockroaches (or, yes, bedbugs). Which is correct? Well, all or none, depending on your point of view.
Interestingly, though, no one ever seems to refer to butterflies as bugs, though they're insects, and I don't know of anyone who considers lightning bugs to be pests.
Several words for bodies of water changed meanings between the old country and the new. In England a pond is artificial, but in America it is natural. Creek in British English refers to an inlet from the sea, while in American English it describes a tributary of a river.
Eh... sometimes. Sort of. In common usage, a "creek" is like a stream or brook: a (mostly) permanent, small flow of water in a channel. Here in the South, people sometimes call it a "crick," which is usually down in the "holler." But if you look at a map of the Chesapeake Bay, especially on the Virginia side but also sometimes in Maryland, you'll see a lot of tidal estuaries called creeks. These were (I suspect, though again, not a lot of fact-checking on my end) likely named by John Smith, who was, of course, English, so I suspect he was using the "inlet from the sea" meaning. Many of them have local names followed by "creek." One example is Potomac Creek, in Virginia, which wasn't named for the river it connects to; rather, both were named for the native group, part of the Powhatan Confederacy, whose main village was located on its banks.
This usage can confuse people who aren't from the area. "That's a 'creek?' It looks like a bay!" Well, it is. Both. You get into categorization problems, like with mountains vs. hills, seas vs. lakes, or what makes a world a planet. Especially with the Chesapeake Bay, which is really the flooded lowlands of the Susquehanna River, which became tidal the last time a bunch of glaciers melted to raise sea level.
Whatever. I spent my childhood in the area, so I probably think about it more than most. Back to the article.
An English watershed is a line or ridge separating the waters that flow into different drainage areas, but in America it’s a slope down which the water flows, or the catchment area of a river.
And that was part of my career, so I always used it in the "catchment area" sense; the ridge that separates watersheds, we simply called the watershed boundary.
English speakers also adopted words from other colonial countries. The language that influenced early American English the most is Dutch.
You see that mostly in the New York and New Jersey area. My favorite, though, is the Murderkill River, in Delaware, which has nothing to do with murders; it's from Dutch words meaning "mother" and "river," so its name, translated, means something like "mother river river."
It's actually a creek.
Bushwhacker, from a Dutch term meaning forest keeper, made its first appearance in print in 1809, in Washington Irving’s comic novel A History of New York, written under the pseudonym Diedrich Knickerbocker. He describes a gathering of prominent Dutch settlers as “gallant bushwhackers and hunters of raccoons by moonlight.”
I will take this opportunity to point out that "raccoon" wasn't Dutch, but came from the Powhatans' Algonquian language. They also gave us "opossum."
Yankee is also almost certainly a Dutch contribution.
And we have yet another thing for which we can blame the Dutch. |
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