Complex Numbers (Book)

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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.

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

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QPdoll

about 13 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

about 16 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

2 Like

September 25, 2023 at 11:35am

#1056196

Gud Speling

Some people peak in childhood. Some of us never peak at all. Here's a peek.

What Happens to Spelling Bee Champions When They Grow Old?
Spelling contests have remained a feature in American life since the Puritans landed on Plymouth Rock — but is peaking at 12 years old all it’s cracked up to be?

Yeah, just to be clear, I'm all for stretching facts to comedic ends, but that bit about Plymouth Rock is plain bullshit with no humor in sight. Spelling was pretty fluid until the early 19th century, and the earliest mention of a spelling contest in the US comes from the first decade of that century.

Anyway. Thinly veiled colonialism aside, the article:

Sixty-two-year-old Brad Williams remembers what he ate for breakfast and lunch — Corn Flakes and hamburgers, respectively — on May 3, 1969, the day he won the Wisconsin state spelling bee after spelling grisaille and lamprey correctly. He also, miraculously, remembers exactly what he was doing on my birthday, September 2, 1978.

See, now, that's the opposite of me, almost. I couldn't tell you what I had for lunch last Thursday, let alone on September 2, 1978. I'm told I was alive then; that's about all I know.

And yet, I'm pretty good with spelling. Not perfect, and I'd be stumped by grisaille because I don't even know what that is but I'm pretty sure it's French.

Williams credits his spelling prowess in large part to the hyperthymesia, since if the word in question happened to be on a study list, he could instantly visualize it on the page.

I'd credit a spelling bee loss to hyperthymesia. If that was one of the words.

As an aside, one snippet of memory I do have from the late 70s involves a spelling competition. Just a minor one, classroom only, and I don't even remember which grade I was in, but it was at least 4th. One girl was asked to spell "dough" and she said d-o-h. That probably stuck in my mind because it was the moment I realized I was better than everyone else. Then, probably a few days later, some bullies beat the shit out of me and I realized maybe I wasn't, after all.

The 4-foot-5 kid had a trophy nearly half his size in his bedroom and considers winning the state bee as one of his life’s major accomplishments.

That's fair. One of my life's major accomplishments was managing to wake up early enough for a job interview, once.

It was invariably a part of Colonial education throughout the 1700s and 1800s, he writes, and the term “bee” referred to many different social events — e.g., “quilting bee,” “barn-raising bee” and “corn-husking bee” — in which an entire community came together, like bees in a hive, for a common goal.

Again, I have some doubts about the practice before 1800. But I'm including this bit to illustrate what "bee" means in this context.

This, though, is why I'm featuring the article in the first place: I'm a sucker for Mark Twain quotes, and apparently, he opened a local spelling bee in Connecticut in 1875:

“Some people have an idea that correct spelling can be taught, and taught to anybody,” Twain remarked. “That is a mistake. The spelling faculty is born in man, like poetry, music and art. It is a gift; it is a talent. People who have this talent in high degree need only see a word once in print and it is forever photographed upon their memory. They cannot forget it. People who haven’t it must be content to spell more or less like thunder, and expect to splinter the dictionary wherever the orthographic lightning happens to strike.”

Another of my misunderstandings, early in life, was that because I was pretty good (not perfect) at spelling and punctuation, I had everything I needed to be a writer. Boy, was I wrong.

The first national spelling bee was held 25 years later in Cleveland on June 29, 1908.

I was also pretty good at math. For instance, 1908 is 33 years later than 1875.

A 500-boy choir sang, three bands performed and one newspaper enthused that “thousands of electric lights will furnish illumination.” Marie Bolden, a 14-year-old African-American daughter of a Cleveland mail carrier, won the competition with a perfect score. Perhaps due to the shock of a black contestant besting all of her white competitors, Maguire suggests the next national spelling bee wasn’t organized until 1925.

This would be funny if it weren't for the goddamned racism. With probably a dose of sexism thrown in for good measure.

As for whether spelling is still an important skill (the answer is yes), the article goes into that, then notes:

Also, cautions Tracey Sturgal, a linguist professor and director of business communication at Marquette University, spell-check isn’t perfect. “People still have to have a certain level of spelling competence,” she explains. “The number one spelling error I get in college papers is when students flip definitely with defiantly, which is surely a spell-check error.”

Spell-checkers also can't tell its from it's, one of my markers when privately assessing someone else's intelligence. Others include your vs. you're, and there, they're, their.

I have to cut this short because my cat has a vet (that's veterinarian, not veteran) appointment, but in brief: unlike child actors in movies and television, who often seem to burn out and/or start using drugs (with notable exceptions like the kid who played Charlie Bucket in Wonka, who pursued a career as a vet (the first kind)), it seems aging former spelling bee champions mostly have their shit together.

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