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 14, 2020 at 12:06am

#975488

Je Suis Content

Well, here's Valentine's Day.

My plan is to spend the evening with my one true love: beer. I'll be doing this at a taphouse (or maybe two), in hopes of eavesdropping on a horrible Valentine's Day couples breakup. I'm easily amused and have a penchant for schadenfreude. Anyway, this means I can't swear I'll do my usual early-morning update tomorrow.

Meanwhile, I found an interesting article about some differences between Anglic and French cultures. It also seems oddly appropriate for Valentine's Day.

http://www.bbc.com/travel/story/20181104-why-the-french-dont-show-excitement

Why the French don't show excitement
Not only is ‘Je suis excité’ not the appropriate way to convey excitement in French, but there seems to be no real way to express it at all.

By Emily Monaco

From the name alone, I'm just going to trust her observations of the differences in cultural approaches.

“You Americans,” he said, “live in the faire [to do]. The avoir [to have]. In France, we live in the être [to be].”

Infinitives as cultural signposts. I like it.

I knew before moving that the French word ‘excité’ was verboten. It is one of the first ‘false friends’ that a student of the language becomes aware of. Most French learners can recall the day that a classmate first uttered the phrase ‘Je suis excité’ (which literally translates as ‘I am excited’) only to have their teacher hem and haw uncomfortably before explaining that the word excité doesn’t signal emotional but rather physical excitement. A better translation of the phrase Je suis excité into English would be ‘I am aroused’.

See, that's exactly the kind of thing I need to know before I attempt to ply my halting French on someone from a Francophone country.

Anglophones, meanwhile, blessed with both words, are free to use ‘excited’ as we please – which we (particularly Americans) do with reckless abandon. We’re excited for our weekend plans, for the summer holiday, to get home after a long day of work and relax in front of our favourite Netflix show.

I'm not sure I've ever constructed the sentence "I am excited" in casual conversation. Even in English, it leaves open the possibility of a double entendre, much like "I'm coming" or "It's not hard." With the crowd I hang out with, I'm careful not to leave myself open to that sort of thing, nor to pronounce the name of the planet that orbits between Saturn and Neptune. Some jokes are just too easy; I prefer to make people work for their puns.

Sadly, this means that I have actually uttered the sentence, "I'm totally psyched." And then I wince inwardly every time.

“I usually say ‘Je suis heureuse’ [‘I’m happy’] or ‘J’ai hâte de’ [‘I’m looking forward to’],” one bilingual friend said. Neither quite captures the intensity of excitement, but it seems these are the best substitutes that French has to offer.

In the course of learning French so far, I've encountered a few synonyms, or apparent synonyms. For example (par exemple), "student" can be translated into French as, at a minimum, étudiant or élève (with variations for gender and number). I suspect that there's a difference, subtle or not, between the two, but I haven't figured out what it is, yet. Older student / younger student, maybe, as with the English "pupil?" Anyway, point is, the first word I learned for "happy" was content (pronounced in the French way with the swallowed 'n's). The second was heureux (f. heureuse as above). Again, I don't know what the actual difference is, but it seems to me that the french "Je suis content" expresses something more similar to the English "I am content," regardless of the overenthusiastic helpful illustrations on Duolingo. Being content isn't really the same as being happy, but then, as far as I know, the French never put "pursuit of contentedness" into the founding documents of their republique.

So, what I'm getting at is, I don't know why the author left that particular synonym out of her discussion.

“If you’re too happy in French, we’re kind of wondering what’s wrong with you,” he said. “But in English, that’s not true.”

I have a suspicion that many French spend their ample leisure time wondering what's wrong with Anglophones in general, even while contentedly selling us their tasty cheeses and delicious wines.

Indeed, those who are unable to show the proper emotional detachment within French society can even be perceived as being somehow deranged...

You might say that they think they're...

*puts on sunglasses*

in Seine.

When we were first dating, my husband used to watch me buzzing around like a busy bee, making plans for the future. He, meanwhile, was able to find not excitement, but contentment, in nearly everything.

There's that word - the English version, anyway.

His frequent motto, whether we were drinking rosé in the sunshine or just sitting in a park, was: “on est bien, là” – we are good, here.

You know, I might just have to steal that for myself, especially when I'm three beers into a bender. Like maybe tonight.

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