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 5 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 18 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

about 21 hours 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

about 23 hours 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

March 7, 2020 at 12:45am

#977380

For A Reason

Today's entry is #2 in a series of 8

assuming I don't fuck this up
for:

Journalistic Intentions

(18+)
This is for the journal keeping types that come to PLAY! New round starts July 1!
#2213121 by Elisa: Middle Aged Stik

Just as with my usual link entries, I'm picking these at random. Little did I know how many of my buttons it would push. I probably should have known; I've been friends with Elisa: Middle Aged Stik

for a long time, and we're often on similar wavelengths.

In case you don't want to play the above video - and I wouldn't blame you one bit - here's my executive summary as I'm watching it for the first time.

1) Chick is living her perfect life.
2) Chick gets diagnosed with nasty cancer.
3) Some kind of aside into Prosperity Gospel: "Nothing is impossible if you believe." Calls it "theology."
4) "Spiritual guarantees." Snort.
5) "It is extremely difficult to avoid falling into the trap of believing that virtue and success go hand in hand."
6) In re #2, "Why is this happening to me?"
7) "It was a mindset that served me well... until it didn't."
8) Questions that boil down to "why do bad things happen to good people," specifically "but I'm a good person, so why did bad shit happen to me?"
9) "Everything happens for a reason."
10) "Near Death Experience Research Foundation"

The Real American Dream

I get the impression that a lot of people, when they hear the words "American Dream," think of the suburban house, two cars, 2.1 kids, dog, that sort of thing - the material stuff. But this speaker touches tangentially on what the American Dream really is, and why it can suddenly turn into the American Nightmare: the idea that, if you work hard enough and follow The Rules, you will prosper. This is so embedded in our culture that it's like when you live in a house where something smells bad - eventually you don't smell it any more, until you go away for a bit, and then you come back and you're like "Peeeew."

If hard work was all it took to get rich, then sharecroppers would be millionaires.

Here's the dark side of it: If working hard leads to prosperity, then if you're not prosperous, well, you must not have worked hard enough. Or you did something else wrong. Or you don't believe the right things. Whatever the case, clearly you're only poor because you deserve to be poor, so why should I do anything about it?

This is, of course, bullshit. People get rich by playing the lottery, guessing right in the stock market, or because of their connections (yes, and sometimes by working hard and smart). Lots of other people stay poor by playing the lottery, guessing wrong in the stock market, or because of their lack of connections, and you never hear the stories of people who worked hard and smart and still ended up with nothing.

Theodicy

The question, "Why do bad things happen to good people?" and its inverse, "Why do good things happen to bad people?" are known in theological philosophy as theodicy.

Essentially, as seen in Wikipedia:

As defined by Alvin Plantinga, theodicy is the "answer to the question of why God permits evil". Theodicy is defined as a theological construct that attempts to vindicate God in response to the evidential problem of evil that seems inconsistent with the existence of an omnipotent and omnibenevolent deity.

A great deal of entropy has been generated by people thinking about these questions, including, apparently, the speaker in this video. Whereas the only reasonable response is to remember that your basic premise is incorrect, and lots of times, random shit just happens. In this worldview, there's no need to appeal to a Driving Force of the Universe. A simple hypothesis that satisfies Occam's Razor, and frees the mind to concentrate on less abstract matters, like, say, how to stop the spread of an epidemic. To use a purely hypothetical example.

Recently, a tornado blew through a town in Tennessee, killing about two dozen people. Someone found an intact Bible in the wreckage, and Franklin Graham crowed something like "This is proof of God's love for us!"

Presumably the same God that just killed two dozen people with fucking wind.

I can't be arsed to look it up, but it was one of the Greek philosophers who noted something along the lines of, either God can do something about evil and chooses not to, in which case he is evil, or he cannot do something about evil, in which case he's not a worthwhile god to worship.

By "evil" above, I'm generalizing to "bad things." When I say "evil," usually I mean shit that people deliberately do, like committing rape, kicking puppies, or failing to tip a restaurant server.

Okay, so what?

I don't want you to get the idea that I lack compassion for this speaker. Cancer sucks. It sucks if you're old, and it sucks if you're young. It sucks if you're alone, and it sucks if you have a spouse and a kid. It sucks, and I don't wish it on anyone (though I come close with Franklin Graham and his ilk). I'm glad she found the strength to get through it.

Listening to her, though, there's something that she's very loudly not saying. Lots of people helped her. Medical professionals, directly. Scientists, indirectly, by coming up with the treatments that seem to be working to save her life. Her fans and well-wishers, who send her letters and emails and whatnot. Whoever set her up as a TED speaker. Her editors and readers. And I only listened through this once, so perhaps I missed it, but I didn't hear even one time when she acknowledged that it was because of all of these humans that she could be standing on stage delivering motivational talks.

Not gods. Humans.

This is what we do. We help. If you're of a religious bent, you can argue that it's some spiritual drive to do so, but I say it doesn't matter - we choose to help (with, of course, a few notable exceptions).

I don't begrudge her what comfort she can draw from her support system, her paycheck, her beliefs. Whatever works. She's probably happier for it, and that's great for her

But the reason she's alive is science.

3 Like • Comment

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