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 3 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 16 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 19 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 21 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

April 2, 2020 at 12:14am

#979969

All Wet

There was a Shyamalan movie a while back, Signs, that was all about (spoiler here) aliens whose only weakness was water. Everybody (except, apparently, Syamalan) wondered why aliens would even bother visiting a planet whose surface is mostly covered in the stuff, and where it regularly falls out of the sky. Decent movie, stupid premise. And don't get me started on what the Wicked Witch of the West must have smelled like if all it took to melt her was water.

Anyway, the point is, water's kind of important for us.

http://nautil.us/blog/-why-water-is-weird

Why Water Is Weird

Water can appear to be “fine-tuned” for life.

Well, duh. Anthropic principle: if it weren't, we wouldn't be here to gawk at it.

In their 2018 study, Hajime Tanaka, John Russo, and Kenji Akahane—all researchers in the Department of Fundamental Engineering at the University of Tokyo, in Japan—tried to tease apart what makes water unique among liquids.

Whatever the technical reason for it, it is unique as far as we know. One weirdness that they touch on in this article: ice is less dense than water, so it floats. What the author doesn't mention is how this protects life in the water; as the ice forms a barrier, the liquid water underneath becomes more temperature-stable, keeping aquatic organisms from freezing (usually). This probably helped evolution along.

The ancient Greeks thought water was one of the four “essential” elements, the others being earth, air, and fire. Homeopathy, which purports to cure illness using small doses of disease-causing substances dissolved in water, evolved out of this, Saykally said. But there are more modern magical claims about so-called “structured” or “hexagonal” water. Some “wellness” practitioners claim humans age in part because we don’t replenish our stock of structured water. Depending on water’s structure, they say, it can penetrate your cell walls more effectively and has all kinds of health benefits.

Well, water does have all kinds of health benefits. As the article points out, though, "structured water" is concentrated bullshit.

The ancient Greeks may have been wrong about water being an essential element, but Saykally says it’s no coincidence that water is essential for life on Earth.

I wouldn't say "wrong," exactly. Just that this was the state of knowledge at the time. Aristotle was wrong a lot, too, but we still study his stuff because it helped form the foundation for modern thought.

Anyway, some people speculate that other liquids might help form life on other worlds. Methane, for example. But anyone positing such a thing is going to have to explain how methane, which has rotational symmetry, can do all the fun things that water, which does not, does. I'm not saying it's impossible, of course; it's a big universe. But I don't think it's very likely.

Maybe there are sentient aliens that are based on ethanol instead of water. I wonder if they get pulled over if they have too much water in their bloodstream...

5 Like • Comment • View 3 comments

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