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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Annette in France

about 7 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 17 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

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

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

September 11, 2019 at 12:04am

#965984

Idols

PROMPT September 11th

Today’s prompt comes from tah20! (With additions by me

)

Write about someone who truly moves/touches/inspires you. Do you know them personally, or admire them from afar? What do they do that makes them special?

Moves? Touches? Inspires? A Jedi craves not these things.

In considering this prompt, I racked my brains to come up with someone I knew. Each of my friends inspires me in his or her own way, but no single one really fits what I feel to be the spirit of the prompt. The closest fit is one I've mentioned before, the one I usually write for, who reads my stuff and is still my friend anyway. That's true friendship, right there - it would drive most people away.

So we're left with celebrities and the like: authors, directors, musicians, etc. - people I don't personally know but have had a great impact on my life.

I'm assuming, further, that I'm limited to living people here. Plenty of people who have died have shaped what I am and what I do. Leonard Cohen comes to mind, there; I first got into writing in a pathetic attempt to emulate his poetry. He was an established poet before he became a singer/songwriter/performer, and his songs reflect this background. He was one of the casualties of the dark year 2016.

That's the trouble with getting old. All your heroes start to die off and you're left holding the void. At least they've left a legacy.

I'm going to have to go with another singer/songwriter/performer, Bruce Springsteen. If you only know his work from the "hits," you're only seeing one facet of his talent, and you probably won't understand this. That's okay; it's a very personal thing with me, something I even hesitate to share here, in this context. But there's no escaping the history; from the moment I first heard Born to Run crackling from a shitty AM radio in a shitty Buick back in the 70s, I've been a fan.

It's not something I can readily explain, either. It's not like I fit into the categories of people he writes about. I tried to, long ago, but it's just not my life. There's a combination of things: the poetic lyrics, the music (which can as easily be droning and lost as it can be energetic and uplifting), the story - and there is a story, about someone who knew what he wanted and then reached out and took it, mostly on his own terms. I've never done that, myself, but I can admire it.

And wouldn't you know, I pick two people, one dead, one still with us, who are known to have achieved greatness in spite of - or, perhaps, because of? - crippling depression. That probably says something about me that I'd rather leave unexamined, for now.

This sort of thing is almost a one-way street, though. I can show my support by consuming their product, along with thousands or millions of other people, but I know I can never impact their lives, by myself, in the way they impacted mine. I'm not one for celebrity gossip, or hero-worship; they're just people, with the only difference between them and us being that they're well-known. And maybe a certain drive that I lack.

All this is not meant to diminish the impact on me of hundreds of other people - including my parents, who raised a stranger's child as their own. That takes a dedication that I certainly don't have. Yesterday was an anniversary of my father's birth, so I thought about what he was like at my age, when I was very young. I'm more like him than I care to admit, most of the time.

But he never did like Springsteen.

2 Like • Comment • View 3 comments

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