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:

Wordsmitty ✍️

about 11 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 14 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 16 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

Lilli Munster 🧿 ☕ 🎃

Sep 24, 2024 at 9:10am
In response to "All Out of Louvre"

Have a wonderful time in France. Have you run into Annette in France yet? She's there too!

2 Like

June 14, 2020 at 12:16am

#985608

Oh, Pinions

"The problem with internet quotes is that you can't always depend on their accuracy" ~Abraham Lincoln

https://qz.com/1513176/john-stuart-mills-philosophy-shows-arguing-online-is-futi...

150 years ago, a philosopher showed why it’s pointless to start arguments on the internet

Wildly inaccurate facts and spurious arguments are unavoidable features of social media. Yet no matter how infuriatingly wrong someone is, or just how much counter-evidence you have at your disposal, starting arguments on the internet rarely gets anyone to change their mind. Nearly a century-and-a-half ago, British philosopher John Stuart Mill explained, in a few clear sentences, why certain arguments simply won’t go anywhere.

So despite the sensationalist headline, a philosopher made a point about arguments in general, and, quelle surprise, the point applies to arguments on a specific medium.

Philosophers, of course, are often wrong, but it's hard to argue with John Stuart Mill. Because he's dead.

Mill highlights the often overlooked reality that many opinions aren’t based on facts at all, but feelings. And so, contradictory points of information don’t shift emotionally rooted arguments, but only cause people to dig deeper into their emotions to hold onto those views.

Of course, MY opinions are entirely based on rationality and facts, while THEIRS are all emotionally based and WRONG.

And chartered psychologist Rob Yeung, whose book How to Stand Out emphasizes the effectiveness of emotions, rather than logic, in convincing others to agree with you, points to research showing that use of metaphors motivate people to make decisions.

1. What the neurological fuck is a "chartered psychologist?"
2. I've been saying that metaphors are powerful, but does anyone listen? No. They're brick walls.
3. "motivate" points to "use" and thus should be "motivates." Does no one edit anymore? Pay no attention to these editor-less blog entries.

There are often multiple ways of interpreting a single point of information and so, much though some people might like to think they’re right about everything, there are surprisingly few issues to which there’s an unequivocally correct opinion.

Except mine, of course.

Instead of seeking to convince others, we can be open to changing our own minds, and seek out information that contradicts our own steadfast point of view.

Now why would I go and do that? It might turn out that I'm wrong, which of course is impossible, and even if it were possible I could never admit it. So it must be that I've been brainwashed.

Maybe it’ll turn out that those who disagree with you actually have a solid grasp of the facts.

Unpossible.

There’s a slight possibility that, after all, you’re the one who’s wrong.

No, YOU are wrong, and probably a sheep-shagger too.

Okay, seriously, though, like most philosophy, this is not helping. Joking aside, yes, I could be wrong. But am I supposed to think that I might be wrong about issues like:

Science is the best tool we have for figuring things out, even if it's not perfect;

Racism is bad;

Vaccination saves lives;

Apollo 11 put two dudes on the actual Moon;

The Earth is roughly spherical

etc.?

I've run into arguments against all of these positions. Some of them, of course, come from trolls who are just trying to get a reaction, which I try not to provide. But some people actually believe the opposite of the above statments, and they're painfully, objectively, dangerously, wrong.

I also have opinions that, yes, could be wrong. Not only that, but I might have facts wrong. If so, I try to be open-minded when I get new facts. My problem in that regard is that I have a lousy memory and I tend to forget the stupid shit I used to believe. It's one reason I blog. But then I'd have to go back through old entries and cringe at old beliefs, and that's painful so I don't do it.

Still, I understand and respect that some people believe differently than me on, for example, the existence (or lack thereof) of a deity, or whether or not we're living in an advanced simulation (or that, effectively, these are the same belief). I admit that if a flying saucer landed near me and a little bald guy came out and asked me to take him to my leader, well, first of all, I'd ask if he could wait until we have a sane leader, but mostly, assuming I'm sober, I'd abandon my disbelief in sentient extraterrestrial life in a snap.

Unlike some people on the internet, I do seek out contradicting views, or, at the very least, I don't automatically dismiss them -- unless they're patently absurd, like the contradicting views to my bullet points above.

Still, nothing's worth the drama or hassle of arguing about it on the internet, or even, usually, in person. Just have a beer and relax already.

1 Like • Comment • View 2 comments

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