Complex Numbers (Book)

Blog Calendar

◄ September ► 2023
S	M	T	W	T	F	S
					1	2
3	4	5	6	7	8	9
10	11	12	13	14	15	16
17	18	19	20	21	22	23
24	25	26	27	28	29	30
Archive				RSS

About This Author

Contact Me

Come closer.

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.

http://www.ddewey.net/mandelbrot/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

September 28, 2023 at 10:04am

#1056335

This Side of the Pond

England invented English. Here in the US, we perfected it.

The Early Days of American English
How English words evolved on a foreign continent.

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.

7 Like • Comment • View 2 comments

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.