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 28, 2020 at 7:19am

#986685

Protein or Amateurtein?

Since Avatar: The Last Airbender showed up on Netflix recently, I finally got a chance to watch the series. I don't have the attention span to really binge it, so it's taken me this long to finish it. In honor of reaching the final episode last night, I also finished a bottle of gin, with the (in hindsight) obvious effect of making it so I have no idea what happened in the final episode. The... good guys... won? I think? Oh, sorry if that spoils it for someone. Also, Darth Vader was his sled and Rosebud was Luke's father.

So now I have to watch that episode again, sober.

All of this is really just my explanation for why I'm posting this at 7 in the morning, after sleeping it off, rather than at midnight, when I was barely able to see the screen, let alone type coherently.

https://www.space.com/possible-extraterrestrial-protein-meteorite.html

First known extraterrestrial protein possibly spotted in meteorite

But the find is preliminary.

Yeah, hedging like that really isn't going to stop people from going "Hey look they found life in outer space!"

A research team claimed to have found the first known extraterrestrial protein, spotting it in a space rock that fell to Earth 30 years ago.

So I can't stress this enough: while proteins are essential for life as we know it, the presence of a protein doesn't imply even the most basic life, let alone Vulcans or Gallifreyans. No, the article doesn't mention anything like that, but every time I see something about "astronomers find earth-like exoplanet" there's always that leap to "Klingons!" It's as predictable as the snickers whenever someone names the seventh planet of our solar system.

The protein, which the researchers proposed calling "hemolithin," consists of chains of the amino acids glycine and hydroxyglycine, as well as iron, oxygen and lithium atoms, the researchers reported. And the hemolithin is significantly enriched in deuterium, a heavy isotope of hydrogen that features a neutron in its atomic nucleus along with a proton.

I can't be arsed to look for this now, being too hung over and all, but I vaguely remember something about astronomers finding evidence of hydrocarbons in outer space. Like, via spectroscopy. I'm certain that, at the very least, they found a cloud that's mostly ethanol -- because, obviously, that's something I'd remember.

At the moment, the distinction between a hydrocarbon and a carbohydrate escapes me, mostly because of the effects of one or the other going on in my body right now (by which I mean, a hangover). But the point is, carbon is really very abundant in space dust, and it forms molecules very easily (hence why there's a whole branch of chemistry devoted entirely to carbon molecules). I guess what I'm trying to say is: it should surprise exactly nobody that there are long-chain carbon molecules in outer space. Being science, though, it's important to actually find them and not just assume that they exist. It's also important to actually find them and not just, for example, end up with Earthly organic contamination on a 30 year old meteorite.

I mean, the deuterium seems promising in that regard, but I don't know enough to know if there might be other explanations for it.

Such deuterium enrichment does not appear in amino acids that form here on Earth, and so it "indicates protosolar disc or molecular cloud origin," McGeoch and his colleagues wrote in the paper...

And that seems like a leap to me.

To be clear, the researchers aren't claiming that the hemolithin is evidence of alien life. But the discovery, if it holds up, would still be of great interest to astrobiologists. That's because it would show that very complex chemistry of the sort employed by life as we know it on Earth occurs of its own accord in deep space.

Considering that Earth is part of "space," again, it wouldn't surprise me in the slightest; if it happened here, it could happen other places. I think the important question is whether long-chain carbon molecules can form and survive things like the solar wind and cosmic rays, which our atmosphere and magnetosphere generally protect us from.

The implication would be that it doesn't take a miracle for bona fide biochemical systems to get up and running — and that, therefore, life might be widespread throughout the cosmos.

This idea might annoy people of a certain religious bent, but it tracks with what I've been suggesting for years: that life, simple life anyway, is probably common in the cosmos. What I question is the existence of sentient life in our vicinity. Evolution in no way requires the development of anything like what we consider "intelligence." There are other ways for organisms to survive and thrive, as is evidenced by the millions of species of plants, insects, bacteria, fungi, etc. here on Earth, all of which have been evolving for exactly as long as we humans have and yet have, thus far, shown little indication of building spaceships or broadcasting radio. Unless, of course, you subscribe to the idea that we're actually puppets of the bacteria, which is probably a stretch.

Mostly I'm putting this out here because there still seems to be an unwarranted cognitive leap in popular thinking from "life" to "space aliens that we could potentially communicate with."

But such talk of implications is preliminary; the new paper is still under peer review, after all. And not everyone is onboard with the results.

This is a good thing. It's important to be skeptical about things like this. However...

"The main problem is the occurrence of hydroxyglycine, which, to my knowledge, has never before been reported in meteorites or in prebiotic experiments. Nor is it found in any proteins," Bada, whose research interests include the geochemistry of amino acids, organic cosmogeochemistry and the search for alien life, told Space.com via email. "Thus, this amino acid is a strange one to find in a meteorite, and I am highly suspicious of the results."

...just because something's never been found before doesn't mean it's impossible. After all, isn't that was "discovery" means?

Also, I have to say that "cosmogeochemistry" sounds to me like an abomination of a word.

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