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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Elycia Lee ☮

about 3 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

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Wordsmitty ✍️

about 16 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

about 19 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

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Elisa: Middle Aged Stik

about 21 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

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Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

April 21, 2020 at 12:02am

#981659

Framed

I'm sharing this link, not to get embroiled in a political argument, but because this has implications for all writing.

https://www.fastcompany.com/90208548/this-is-how-tiny-changes-in-words-impacts-t...

This is how tiny changes in words you hear impacts your thinking
In a fascinating look at language, a Professor George Lakoff lays out how political parties can sway supporters with tiny tweaks in word choice.

It's not just political parties, of course; it's generally applicable to writing (including writing for video).

In 1973, America watched as then President Richard Nixon vehemently declared on national television, “I am not a crook” in regards to the Watergate scandal.

Not many people believed him.

Partly it's because of his word choices. Partly it was because everyone knew he was a crook and voted him in anyway.

The major mistake Nixon made was in his framing.

While I know what this author is trying to say here, I think we can all identify what Nixon's "major mistake" was, even if you're too young to remember Tricky Dick.

George Lakoff, a professor in cognitive science and linguistics at University of California, Berkeley, makes the point in his book Don’t Think of an Elephant! that when trying to get your point across, refrain from using the other side’s language.

Buy my book, and this guy's book too! Also, for the eleventeenth time, life isn't binary; there's more than two sides, always.

So, exactly what is a frame?

“A frame is a mental structure that is represented in the brain by neural circuitry,” Lakoff explains. Frames shape the way people see the world, and consequently, the goals we seek and the choices that we make.

I'd like to point out that "neural circuitry" is a metaphor. The brain isn't a computer. Also, that's a shitty definition, because it requires drilling deeper with more definitions.

Not that I believe that could define "frame" any better, but I'll take a stab at it, at least from a writing point of view: a frame is a mental metaphorical structure built on word associations, their emotional content, and their implications.

How'd I do?

Think about it this way: Something that has a “95% effective rate” will sell better than something with a “5% failure rate.” It’s all in how you frame it.

I've gotten into the habit of always turning statistics around in my head, anyway. It's a glass-half-full or half-empty kind of thing. Well, as an engineer, I see a glass overdesigned in capacity for its contents, but that's irrelevant. And as a drinker, I see a need for more beer.

Because subtle differences in languages might shape our thoughts and change how we experience reality, learning a new language can activate new frames. Quite simply, speaking differently requires thinking differently.

I stuck this article in my queue months ago and I'm just now getting to it, but I have a dim recollection of wanting to make a point based on this paragraph, something related to me learning French. But obviously, learning a new language hasn't helped my general memory all that much, as far as I can tell. Probably had something to do with this "thinking differently" thing being one of the reasons I decided to try learning a new language.

The rest of the article deals more directly with the political implications, specifically American politics, and that's not something I want to wade into here. It's the general ideas about framing as they relate to writing that I want to emphasize: word choice is important, and we need to consider not only the words but their possible emotional baggage.

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