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 9 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 22 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

a day 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

January 19, 2020 at 12:02am

#973682

Feelin' Good Again

PROMPT January 19th

Write about something ordinary that brought you unexpected joy.

Other peoples' misfortune.

Okay, no, not really. It's true that schadenfreude, when applied to someone who I deem to deserve it, does often make me smile. Unfortunately, more often than not, bad things happen at random to people who probably didn't do anything especially wrong; conversely, people who deserve a dose of instant karma can go indefinitely without any noticeable ill effects. Life, it seems, is fundamentally unfair.

Thing is, it's usually unfair in my favor. I suppose this is what they call "privilege." I've noted in previous entries some misfortunes that have befallen me: health issues, inconstant wives, disappearing friends, dying parents, dead cats, the closing of my favorite bar, etc. But these are ordinary problems, nothing that billions of other people haven't experienced. Yes, they affected me deeply (some more deeply than others; man, I miss that bar), but they're all part of the general human condition.

On the upside, I've been really fortunate in other ways. For instance, I've only been in a formal job-interview situation twice in my life (I've had more than two jobs, but those were more "Can you show up on time and do the work?" "Yes." "Okay, you start tomorrow.") [Narrator: he never could manage to show up on time.] On the more recent of those occasions, the custom of throwing curveball questions at the applicant was just getting into full swing, and one of the interviewers, a woman with a can-I-speak-to-your-manager haircut, broadsided me with the question, "What brings you joy?"

I stammered out something about seeing something I designed becoming reality - true enough, but boring - and I didn't get the job. That's okay; I'd just been checking out the hue of the trans-fence grass, and I didn't really need to switch jobs at the time. I don't know what answer she was expecting, though. "Hookers and blow?" "Crushing my enemies under my Reeboks?" "Really good sex?" "Getting cabbaged and inflicting karaoke on strangers?"

Still, that question stuck in my memory for some reason, prompting me to identify actual joy on those occasions when I feel it - so something good came out of the failed interview, after all. It's important, especially to those of us prone to depression, to take our joy where we can, but to do that we have to know what it feels like.

There was the time, driving on a deserted highway in central Virginia on a quiet Christmas Eve, when the full moon silvered the empty landscape and limned the mountains in its pale light.

More recently, and I think I noted this one in a blog entry last year, there was the occasion when I was about to see Bob Dylan perform live; I had just achieved beerenity at a bar in NYC when Counting Crows' "Mr. Jones" issued from the bar's sound system (it's CC's tribute to Bob Dylan, so it seemed absolutely serendipitous).

Another time, toward the end of an excellent Springsteen concert, all the lights in the stadium came on at once and the band launched into "Born to Run," and it was the perfect climax to an awesome four-hour show, like a cool breeze after a long, hot, but productive day.

Come to think of it, most (but not all) of my peak moments involve alcohol, music, or (preferably) both.

But - and I think I've mentioned this here before also - it's almost always the more conventionally depressing songs that make me happy, while upbeat, optimistic songs tend to make me stabby. Why that might be is above my pay grade. But the same is true for booze; they tell me it's a depressant, so why does it make me so fucking happy?

Which is why today's musical selection, by Robert Earl Keen, is, on the surface at least, an outlier. A casual reading of the lyrics pegs it as a happy song. "It feels so good, feelin' good again." And yet...

I have a different narrative for it. Now, I'm no expert in music, but it sounds to me like it's presented in a minor key, though a quick glance at the published chords indicates that the body is a G-Bm-Em-D progression, with a C-D-G refrain. I guess the Bm-Em transition gives me that impression. Point is, minor chords are almost never used for happy music, and I stand by my impression, because it's mine.

But how do I reconcile the (to me) inherent sadness of the chord progression with the apparently-joyful lyrics? Is it just because it's the genre "country," which tends toward the despairing and desperate? Here are the full lyrics with chords marked.

Well - again, this is my personal opinion - look at what happens in the lyrics. Everything is coming up Keen: "A chill north wind was blowin' but the spring was comin' on" sounds a lot to me like when I'm coming out of a depressive episode. "Stepped into the hall and saw all my friends were there" - I mean, seriously, how often does it happen that "all" of one's friends are in one place? "My favorite band..." finding unexpected cash to spot the crowd a round of drinks... and then the one person (presumably a life partner or former one, as they are not included in "all my friends" earlier) he's wishing to see shows up.

All of this paints a picture that's relatable to me, even though country isn't my go-to genre: the gathering place, the drinks, the music, the longing that is ultimately fulfilled. But then it hit me: that's because it's not real. I don't mean that in the sense of "of course it's not real; it's just a song." And I'm probably giving this more thought than it deserves, but everything fits when you think of the scene as a dream - or an idealized afterlife. In other words, I think the narrator is dead or dying, and at the end, he's finally getting all of the simple pleasures he'd been hoping for.

Man, I hope Keen doesn't see this analysis, though I'm sure it reveals more about me than it does him. But I'll stand by this much: giving the song that interpretation makes it better, to me. Happier. More joyful. Against all expectations.

And I wanted you to see 'em all
I wished that you were there
I looked across the room
And saw you standin' on the stair
And when I caught your eye
I saw you break into a grin
It feels so good feelin' good again

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