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:

Write_Mikey_Write!

about 19 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 21 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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