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 7 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 20 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 23 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

a day 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

February 11, 2020 at 12:04am

#975252

Terrible Lizards

Anyone ever play the game with Google Translate where you put in something in English to translate to, say, Latin - and then put in the Latin to translate back to English?

The results are sometimes hilarious, and often incomprehensible. Or both.

For example, I put in "These are the times that try men's souls." The Latin it returned was: "Haec tempora quaerunt animas." Okay, my Latin is rustier than an old barge, but that seems about right. But then I put "Haec tempora quaerunt animas" into the source and asked for English.

I got, "This is the time to destroy."

Language, as I'm rediscovering while learning French, isn't usually a one-to-one correspondence of concepts. I mean, I've known this for a long time, but there's a difference between knowing and applying. Like, it's well-known that whereas in English you might say "I'm 33 years old," in French you say something like "I have 33 years." Same idea, different modes of expression, though in that particular example it's pretty easy to tell what is meant.

While comparative linguistics has been interesting to me for a long time, I've usually been too lazy to do more than scratch the surface of other languages. Still, articles like this one usually catch my eye:

https://www.bbc.com/future/article/20200116-what-happens-when-you-have-no-word-f...

What happens if you have no word for 'dinosaur'
English is the world’s dominant scientific language, yet it has no word for the distinctive smell of cockroaches. What happens though, if you have no words for basic scientific terms?

I'm not going to do my usual copy/paste of segments; I'd pretty much have to reformat a lot of words, and I just can't be arsed. But I do have a couple of general comments.

First, comparative linguistics aside, the development of an individual language is interesting in itself. Famously, English isn't shy about borrowing words from other languages, which may be one reason it's so versatile (but also one reason why it sometimes doesn't make any objective sense). In addition to simple words, like "tattoo" (from the Samoan language, apparently), we've adapted countless terms, scientific or otherwise, from ancient languages like Latin or Greek. If a concept comes along and we don't have a word for it, chances are someone will coin something with an ancient-language base; "television," for example. And then we make it our own; in the US by saying "TV," and in England by saying "telly."

Second, these concepts are coming at us fast and hard. Some inventions require their own names. Something like "starship" was easy; it compounds two simple words from English itself. But then someone invented the transistor, and, more importantly, called it the "transistor" instead of something that I can't even imagine being built up from Anglo-Saxon words. Smallswitch, maybe? But that doesn't really convey all the uses of the transistor, which also relies on other concepts the precursors of the English couldn't have known, such as electricity, electrical current, conductivity, electrical resistance, amplification, and so on. German is much better at mashing words together to make new words.

Point is, to get back to what the article was saying, we invented the word "dinosaur" from ancient languages, languages without that symbol-concept, but the original meaning is largely irrelevant to what we think of when we think "dinosaur." So other languages have, at least, two options: they can just borrow the word like English might, at which point you have someone speaking, I dunno, the Navajo language and you can clearly hear the word "dinosaur;" or you can, as the article suggests, come up with an entirely new word for the concept based on words your language already has.

Like I said, I find the whole thing fascinating, and I sometimes wish I had the time and energy to learn even more languages to feel for myself the baffling complexity of the human capacity to assign sound-symbols to concepts.

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