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 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 29, 2020 at 12:06am

#974473

After All

PROMPT January 29

Everyone did a great job with filling the war chest yesterday! There's so many great prompts and I'm sure Em will love every one of them.

Since today *is* War Chest Wednesday, I'm going to grab one.

What is your favorite virtue? Give a few examples like kindness, cleanliness, tact, truth, generosity. Is your favorite one you possess, or one you simply admire in other people? Do you have a strategy to develop it yourself?

Prompt is courtesy of Kåre Enga in Montana

Snark.

Wait, is that a virtue? No? Everyone hates it? Well, shit.

I'm glad Kåre Enga in Montana's other idea, the one involving a certain mouthless feline, didn't make it into the prompt. Still, I find the juxtaposition of "tact" and "truth" amusing, since those are sometimes incompatible.

Other examples of virtues? Well, I suppose there are two loosely defined kinds of virtues: directed toward the self, such as tenacity, self-control, or mental flexibility; and directed toward others, like charity, fairness, and loyalty. There might be a third kind, but it's intertwined with religion, and one does not need to be religion to be virtuous; sometimes, it seems, religion gets in the way.

But there's one virtue that, I think, influences many of these others, both inward and outward, and that is compassion. One can be compassionate to others, obviously; but it's also good to be compassionate to oneself. I know I'm harder on myself than I am on others, and I could stand to cut myself a little slack now and then.

Unfortunately, for myself, compassion isn't something that comes naturally. I have to think about it deliberately. I mean, my first reaction when someone's holding up a line at the grocery store, or cutting me off in traffic, or trying to sell me insurance, is usually annoyance. I have to reach for serenity, and I have to reach for compassion: knowing, logically, that everyone is the star of their own show, and they can each justify their own actions, and maybe they're just having a bad day. Maybe their dog just died, or their house has a leak, or they're worried about a friend.

And yet, compassion is the one thing I want to strive for. On a purely selfish level, it helps me as a writer to try to see others' points of view. But more importantly, I think that if I could crack the code of compassion, perhaps other people will appreciate it if they know someone cares about what they're going through, whatever that may be.

There was some discussion a while back about what people are "really" like. I believe that we have to practice being who we want to become. If something that you want to be isn't a part of your character, is it "fake" to try to practice that virtue until it becomes a part of you? I don't think so, myself; otherwise, how could any of us deliberately change? So, no, I'm not a compassionate person. But I'm working on it.

I'm not very good at it. It's not like I can learn it like I'm learning French, or like I learned how to do engineering. But I've seen peoples' actions that I have identified as "compassionate," and that's what I strive to emulate.

One such person that stands out for me is a lesser-known singer/songwriter named Dar Williams, who as far as I'm concerned is the Avatar of Compassion -- so it's her song I'm going to feature to tie in with this theme.

Go ahead push your luck
Find out how much love the world can hold
Once upon a time I had control and reined my soul in tight

Well the whole truth
Is like the story of a wave unfurled
But I held the evil of the world
So I stopped the tide froze it up from inside
And it felt like
A winter machine that you go through and then
You catch your breath and winter starts again
And everyone else is spring bound

And when I chose to live
There was no joy it's just a line I crossed
It wasn't worth the pain my death would cost
So I was not lost or found
And if I was to sleep

I knew my family had more truth to tell
And so I traveled down a whispering well
To know myself through them...

6 Like • Comment • View 5 comments

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