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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 16 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 18 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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