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:

Annette in France

about 9 hours ago
In response to "It's Hot in The South, Once Again!"

The quote is too small and low contrast to be read.

Elycia Lee ☮

about 19 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 ✍️

a day 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

Sep 25, 2024 at 7:53pm
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

Sep 25, 2024 at 3:20pm
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🎶

Sep 25, 2024 at 2:49pm
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

August 3, 2019 at 12:14am

#963641

Humans.

Today, I'm going to wade into a minefield.

https://www.theatlantic.com/family/archive/2018/06/imagining-a-better-boyhood/56...

In hindsight, our son was gearing up to wear a dress to school for quite some time. For months, he wore dresses—or his purple-and-green mermaid costume—on weekends and after school. Then he began wearing them to sleep in lieu of pajamas, changing out of them after breakfast. Finally, one morning, I brought him his clean pants and shirt, and he looked at me and said, “I’m already dressed.”

Don't care.

He walked the half block to school with a bounce in his step, chest proud. “My friends are going to say dresses aren’t for boys,” he told me casually over his shoulder. “They might,” I agreed. “You can just tell them you are comfortable with yourself and that’s all that matters.”

Beginning to detect the faint, familiar odor of bovine excrement.

When he walked into his classroom, sure enough, one child immediately remarked, “Why are you wearing a dress? Dresses are for girls.” A teacher swiftly and gently shut down the child’s commentary and hugged my son tightly.

Ah, yes... homegrown fertilizer. Teachers don't hug, nowadays. They get investigated and fired for that shit.

One day when my husband dropped him off, he heard a little girl stand up to a naysayer and shout, “Boys can like beautiful things, too!”

Sure he did.

Thing is, though, possibly bullshit anecdotes aside, the author makes, in my opinion, excellent points.

Maybe you weren't expecting me, a middle-aged man, to say that. I'm supposed to be the Enemy here, trying to hold on to antiquated ideas of gender norms in the face of a rethinking of everything that "gender" means.

Fuck that.

Like I said, I don't care - one way or the other.

Thing is, though - I don't always understand. I feel like there's a subtext to every conversation about sex and gender that I'm not a party to; I'm not even invited to the party. Maybe that's because, like I said, I'm a middle-aged man and thus subject to the social expectations thereof - in short, I'm put in a box as much as anyone else is. Maybe it's because, when I was a kid, my parents were progressive enough not to put all those "this is what boys do" crap in my head. Probably it's something else entirely; I don't know, because whatever it is, I'm missing it.

But it doesn't matter. I don't have to understand why some penis-owners insist that they're women, or the other way around; I just have to accept it - or not. I do accept it, because, clearly, it happens. I don't have to understand why anyone does anything, for that matter, and as long as they're not hurting anyone else in the process, I find it difficult to give a shit.

Way I see it, everyone - everyone - has stereotypical "masculine" and stereotypical "feminine" traits, likes, and dislikes, in varying degrees. As with most social things, it's not binary, but on a continuum. For instance, I know a guy whose hobby is collecting vintage Barbie dolls and making little dresses for them. I know another one who likes to enjoy a "Princess Cake" for his birthday every year, complete with pink flower frosting. Not that it matters, but both these people identify and present as men, and they've each been married (to women) way longer than I ever was; I only say this to point out that their other traits are in no way stereotypically effeminate.

So, I don't know what I'm missing about this conversation. People like what they like. To shame them into not liking something is basically making it clear that you don't accept them as a person, to attempt to suppress their expression of creativity or joy or whatever.

And that sucks, no matter who you are.

Therefore, I keep reading about it, and I attempt to have conversations about it, but I feel like every time I bring it up, someone decides I'm not, I dunno, ideologically pure enough to even have the discussion with, and I'm left even more bewildered than before. Just let people be who they are, is my view - again, excluding those who want to do harm; I'm not advocating, say, letting pedophiles have their way. Or, maybe they don't know who they are; in which case, let them explore.

Anyway, yeah, I know I'm rambling. That happens when I can't quite wrap my head around something. Maybe someone else can shed some light on the subject for me.

4 Like • Comment • View 5 comments

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