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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10 Most Recent Blog Comments:

Elycia Lee ☮

about 5 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!

2 Like

Wordsmitty ✍️

about 18 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.

1 Like

Annette in France

about 21 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.

1 Like

Elisa: Middle Aged Stik

about 23 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.

2 Like

Brandiwyn🎶

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

Love it when stars align!

2 Like

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

March 30, 2020 at 12:16am

#979659

Cunning Linguists

One left to do after this... entry #7 of 8 for

Journalistic Intentions

[18+]
This is for the journal keeping types that come to PLAY! New round starts July 1!
by Elisa: Middle Aged Stik

How language shapes the way we think

This harks back to a post I made in the Before Times, way back on March 1: "It Doesn't Count"

. It's something that's long been fascinating to me, so I'm glad this one came up in my random walk through the prompts.

Now, I can't be arsed to look it up, but the idea that language shapes thought has its detractors -- or, at least, that it's as pronounced (pun intended) as this speaker asserts. Me, I'm no expert; I'm just drinking from the fountain here.

Part of my own cognitive problem is that I've never bothered to dive deep into any language except my own. My knowledge of idioms and grammatical constructions is very English-oriented, and American English in particular. I can play with it, twist it, and hopefully get my points across even if I don't hew to standard grammar and phrasing -- but it's definitely English.

What little I've learned of French in this regard has been enlightening. For example, we Yanks might say something like, "I have a headache" or "My head hurts." Well, the equivalent French phrase, strictly translated, would be something akin to "I have bad in head" or "I have wrong in head" (the bad/wrong word being mal). If I told someone I had a wrong in my head in English, they'd assume I need to be committed somewhere. I should note that I'm far from fluent in French, so I could have mal about all of this, but that's what it comes across as at this point in my learning.

Point is, that's a language that, as these things go, isn't all that dissimilar from English. I mean, obviously most words are different, English is less inflected, and the French syntax is often scrambled from an English point of view (ours is scrambled from their point of view), but for the most part, there are parallel parts of speech; things are countable; there's a left and right, up and down; colors are recognizable; and so on.

But that makes me pose a different question: why are there different languages in the first place? Language, obviously, evolves in communities that are separated. In that sense, it's similar to, but much faster than, biological evolution. But language also has a feature not usually found in (eukaryotic) evolution, which is cross-pollination. In biology it's known, as far as I understand it, as "horizontal gene transfer," and apart from genetic engineering, you just don't get a lot of that in nature (for the purists out there, it happens a lot in bacteria and other prokaryotes, but I'm not talking about them). But with language, horizontal meme transfer (using "meme" here in the sense that Dawkins originally coined the word) happens all the time.

Answering that question is above my pay grade, though the one thing I can say with a high degree of certainty is that it wasn't God's punishment for building a fucking ziggurat. (If it were, then why did language keep on evolving long after Babel?)

I can only hypothesize that it was groups of people who separated themselves and started thinking independently of each other. In which case, which is more true: that language shapes thought, or that thought shapes language?

I suspect there's a bit of both going on. Our ability to think in new ways

when learning a new language is, well, very human.

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