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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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 2, 2020 at 12:06am

#972434

Livin' In The Future

PROMPT January 2nd

Yesterday, we looked to the past. Today, we’ll write about the future! (No resolutions, I promise, Robert

)

2030 sounds so futuristic, but it’s only ten years away! What do you think the world will be like ten years from now? What inventions do you think we will have by then? What are some changes you think will happen in the world in the next ten years?

I discovered science fiction young.

Between the TV shows like Star Trek, short stories (which were still a published thing at the time), novels and even comic books, I got to see what I thought were different authors' visions of The Future™.

Later, I came to understand that it is not the job of a science fiction author to foretell the future; it is his or her job to prevent it.

I don't mean the post-apocalyptic wasteland subgenre, or not just that. I mean that science fiction is, essentially, a philosophical form of writing that explores the repercussions of change on humans and their societies. "But not all science fiction is like that," one might object. "Sometimes it's just space empires and adventure stories." This is not the place to get into the fine details of what is and is not science fiction; besides, any such argument starts to wiggle perilously close to the No True Scotsman logical fallacy. Yes, I understand the objection. In rebuttal, I'll just post a link to Sturgeon's Law.

Notably, Theodore Sturgeon was a science fiction author.

Point being, every once in a while you come across a "news" story or fluff piece that explores what this or that science fiction author "got right" and what they "got wrong." While sometimes intriguing, they miss the point; again, SF authors aren't here to foretell the future. On those occasions when they do explicitly foretell the future, the only reason they might have a slightly better track record is because they are, pretty much by definition, interested in the trends of science and technology, and anyone with such interests can prognosticate some advancements to some degree of accuracy.

On the other hand, those are some of my interests as well, and yet I have no idea what things are going to look like in 10 years because it's not just about science and inventions; it's also about human interaction with them.

Let me provide a quick example. It was fairly common, in the 60s and 70s, to predict the advent of videophones. As I recall, the idea was you have the existing telephone line technology, only instead of just having an audio transmission, you also have a video screen. Some enterprising individuals even figured out how to make the things work. But people freaked out. What if someone calls when I'm getting out of the shower? What if my hair is a mess? What if my house in the background is a mess? So, even though we had the technology, in spite of science fiction "predictions," the vidphone never caught on.

At least, not until phones were basically replaced by computers, smartphones and other internet - connected devices, at which point people started Skyping and videoconferencing and whatnot. It was the vidphone, manifest at last - but in a way that I don't think anybody foresaw.

It was a failure of imagination all around. The general public couldn't conceive of a way to make a phone call without the traditional "run to the phone to answer it when it rings" routine. Would-be vidphone makers couldn't come up with a way to change that routine, or otherwise alleviate the public's distrust of the concept. It was only when a lot of other factors came together, both technological and social, that we could have the effective equivalent of the videophone.

I see similar arguments playing out around the idea of self-driving cars, right now. Lots of people are distrustful of them, in the same way lots of people are scared shitless of flying because of the possibility of a plane crash, but think little of speeding to the airport if they're a little late. The chance of death or serious injury is many orders of magnitude higher in the latter case, but people focus on the former possibility.

Right now, in the US, about 40,000 people die each year in auto accidents. That number's fuzzy, but let's use it for arguments' sake. You can also find statistics by miles driven if you look for them; I can't be arsed. Logically, then, if we can switch to self-driving cars and get that number down by at least one or two standard deviations - say to around 32,000 - then autonomous vehicles could be shown to be inherently safer than human drivers, and in a perfect world, we'd just switch over.

But an autonomous vehicle killed one pedestrian, and the consensus became "we can't have these things; they're too dangerous." We're apparently perfectly okay with killing 40,000 of ourselves every year rather than reducing that number by changing the way things work.

People, in short, are fundamentally illogical. And that is why you shouldn't trust anyone who says "this is the way things will be in 2030." Not even me. There's no telling how the public will react to a change, either from without or within.

But I will make one prediction that you should definitely take seriously: People are not going to fundamentally change.

And that's why we continue to need science fiction: to prepare us non-changing meatsacks for the things that do change, be it technology, scientific discoveries, unexpected breakthroughs, alien invasion, the zombie apocalypse, 40 foot sea level rise, nuclear annihilation, or the inevitable takeover of all forms of entertainment and government by Disney.

Don't worry, darlin'
No baby, don't you fret
We're livin' in the future
And none of this has happened yet

7 Like • Comment • View 11 comments

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