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 3, 2020 at 12:04am

#976935

Breathe

I've had a few today. The drafthouse cinema had a screening of Dr. Strangelove, a movie that is all over popular culture and yet, somehow, one that I had never seen.

Now I've seen it, and I understand so much more.

Altered state of mind notwithstanding, I'm still doing one of these entries.

https://www.popsci.com/where-does-fat-go/

When you burn fat, where does it actually go?
Getting rid of fat can be exhausting, but it's as natural as breathing.

One of the basic principles of physics, as I understand it (and I don't pretend I do, much), is that of conservation. Specifically, conservation of mass/energy. I mean, there are other conservation laws: angular momentum, particle charge, whatever. For our purposes today, though, I'm only concerned with mass/energy.

I put the slash in there to emphasize that while the total of mass and energy must be conserved, one can be converted into the other, as per some guy named Albert.

So, in my quest to lose weight, it struck me: where does all that weight actually go? Shit? Piss? Sweat? What? It's not like it can just, as the gymrats put it, "melt away." Weight is mass under the influence of gravity, and it can't just disappear.

Or can it? I mean, again as I understand it, there's nothing inherent about the conservation laws. That is, they don't arise as a logical result of anything basic; they're just so consistently observed that any perceived violation of them would be tantamount to the discovery of unicorns, fairy dust, or honest politicians. Still, their consistency means I wondered about the mechanism by which excess avoirdupois leaves one's body and goes... somewhere else.

My limited amount of Googling on the topic yielded nothing of use, just more stupid motivational quotes which are useless to me because my contrarian nature makes me rebel against anyone who says "YOU CAN DO IT" by going "NO I CAN'T." And then I just happened upon the above article, which leads me to...

Humans have two kinds of fat cells, brown fat cells and white fat cells, the latter of which are far more common.

I first encountered this article when I was sober, and I remember thinking something about that being a metaphor for racism in the US, but the mechanics of that eludes me at the moment. Eventually I'll figure it out again.

So what happens during that shrinking process, when you actually lose some fat? Contrary to popular beliefs, all of the fat used by the body doesn't get turned directly into energy, it doesn't magically transform into muscle, and it doesn't exit with your other solid waste.

I was pretty sure magic wasn't involved, so... thanks, PopSci?

But it has to go somewhere. That's the beauty of the conservation of mass. Much as some people might want it to, fat can't just disappear.

Again, it's more like "conservation of mass/energy." But I can't be arsed to follow the link in the article.

The fat inside your adipose cells is stored and then sent back out into your body as triglycerides. These go through a series of chemical reactions to convert into energy, but the process isn’t 100-percent efficient. Our bodies are good, but they’re not that good.

Well, duh. No conversion process is 100 percent efficient. Some French dude figured that out when he was studying steam engines way back when steam engines were cool. Carnot? Maybe.

Some of the water produced as the body consumes fat exits in the usual way, through your urine and sweat. But a 2014 study in the British Medical Journal found that most of the byproducts of fat (including all that carbon dioxide) leaves the body through the respiratory system.

Great. So what you're saying is, by losing weight, I'm contributing to global warming.

Oh well. Tough shit.

But hey, this gives me an excuse to embed a Pink Floyd video again.

For long you live and high you fly
But only if you ride the tide
And balanced on the biggest wave
You race towards an early grave

2 Like • Comment

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