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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QPdoll

about 9 hours ago
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I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 12 hours ago
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Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

about 23 hours ago
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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
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Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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November 20, 2023 at 9:11am

#1059817

Lucky Town

I'm linking this article from Psychology Today mostly because I appreciate the author's choice of role models.

The Great Resignation of Engagement
5 lessons we can learn from The Boss

By The Boss, he means, of course, Springsteen.

If in doubt that we’ve become less engaged at work, consider a recent Gallup survey of almost 15,000 U.S. employees...

Note to writers: I find that leading off with statistics, polls, or other numerical data is a good way to lose readers. I'd have stopped there if I wasn't promised a Bruce connection, and I'm a numbers guy.

A few weeks ago, I found an answer in the unlikeliest of places: at Circo Massimo in Rome. This answer doesn’t come from the boss in your organization or another. It comes from The Boss—Bruce Springsteen—who my wife and I saw, along with 60,000 other people, at his recent concert near our home in Italy.

Springsteen's maternal grandfather was an Italian immigrant to the US, so I'm willing to bet Italy was a pretty awesome concert. Oh, what am I saying; they're all awesome concerts. I'm way past the "be part of a crowd of 60,000," though. Hell, that's more people than the population of my city.

Here are five lessons I learned from The Boss that I’ve already started applying to become more engaged...

1. Allow your passion to evolve with you. What stood out for me the most at Circo Massimo is that Springsteen is 73 years old and still has his passion.

74 as of this writing. Now, my inner cynic (who takes up most of the space in there) objected to this with "yeah, right, pay me like Springsteen and I'll be passionate." But for once, I squelched the bastard. I'm sure the guy's acquired enough assets to have comfortably retired, and never have to work again, unlike, say, Leonard Cohen, who had to tour in his old age because his financial advisors "lost" all his money for him.

No, remember what I said about luck, yesterday? He knew what he loved early on (music), he possessed the talent and drive to pursue it (I love music, but I have no talent for it), and he was in the right place at the right time to capitalize on that.

Yet, for most of the first two hours, he played more recent songs from his vast catalog, from the past 15 years when the popularity of his albums could not really stand up to the massive response to his earlier work, and not that many people at the concert seemed to be familiar with them.

But his first two albums were commercial flops at the time. When he exploded into world consciousness with Born To Run, that was not the result of evolving passion so much as a deliberate stylistic change to appeal to a wider audience. I remember reading in, I think it was Dave Marsh's biography of Bruce, that he set out to write "the greatest rock and roll album of all time." Once firmly set in fame and fortune, he was able to pursue more personal projects, ones which didn't have the same mass appeal.

Point being, no, I don't think this is good career advice for working stiffs. (What's that, Inner Cynic? Oh, yeah: "Give the people what they want, get rich, and then do what you love.")

2. Don’t go through life—grow through life. “Have you ever seen someone play for three hours like this, without a break?” I asked my wife this as we were driving home from the concert.

Long sets are a hallmark of a Springsteen concert. Three hours means he's definitely slowing down. In the glory days (pun intended) of the 1980s, I attended a couple of 4+ hour shows. Even the concerts I went to in the noughties were quite long, though I didn't time them. Point being, how is that "growth?"

3. Show up as you are. Springsteen did a prelude to his song “Last Man Standing” in which he talked about George Theiss, who was dating his sister and invited him to audition for his band. Bruce went out, at 15 years old, and ended up playing with the Castiles for three years.

I'm not sure this section is very coherent. Also "show up as you are" might work for rock and roll or IT positions; the rest of us have dress codes and such.

The rest of the article continues along the same track.

In terms of role models, we can do worse. I'm just not sure this professor (that's a pun, too, because the author is a university professor, and Springsteen's longtime keyboard player, Roy Bittan, was nicknamed "The Professor," and now that I've explained it, it's lame) is drawing the appropriate conclusions.

We can't all just "do what we love" through life and be successful. If you can, if you have, well... that's luck.

And, lucky for me, it worked for Bruce.

And I had some victory that was just failure in deceit
Now the joke's comin' up through the soles of my feet
I been a long time walking on fortune's cane
Tonight I'm steppin' lightly and I'm feelin' no pain

Well here's to your good looks baby now here's to my health
Here's to the loaded places that we take ourselves
When it comes to luck you make your own
Tonight I got dirt on my hands but I'm building me a new home

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