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:

tj-Merry Mischief Maker

about 21 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

a day ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 28, 2024 at 10:37am

#1080592

Trolley Pop

I've written about the Trolley Problem before. At length. I even wrote a very short story featuring it: "The Trolley Problem"

[18+]. This is very likely to be the last time I feature an article about it; this blog is steadily approaching its end.

The article itself is a few years old, but I'm not aware of any progress in Trolleyproblemology since it came out in 2018. It's also from Slate, so no surprise they got it wrong.

Does the Trolley Problem Have a Problem?
What if your answer to an absurd hypothetical question had no bearing on how you behaved in real life?

It was never meant to have bearing on how you behaved in real life. Consider this article

from Philosophy Now (limited free articles), which concludes:

The answer, in my view, is that there is no definitive solution. Like most philosophical problems, the Trolley Problem is not designed to have a solution. It is, rather, intended to provoke thought, and create an intellectual discourse in which the difficulty of resolving moral dilemmas is appreciated, and our limitations as moral agents are recognized...

I do not believe there will ever be a perfect solution to the Trolley Problem, nor a consensus as to the best possible solution. All we can hope for – and should hope for, as I have argued – is to utilize the tools of philosophy as well as the scientific method to continue this discourse. The Trolley Problem does not have to be resolved; it merely needs to be contemplated, and to be the topic of our conversations from time to time.

That is, of course, the opinion of one philosopher, but it rings true to me. Philosophers, however, aren't known for having a sense of humor. They're all Very Serious Thinkers. We have a different name for philosophers with senses of humor: we call them "comedians." And comedians have been having a field day with various permutations of the Trolley Problem, many of which are legitimately hilarious.

Which is, as the Very Serious Philosopher notes, the point—even if he'd be appalled at the humor elements.

So, back to the Slate article:

I ask because the trolley-problem thought experiment described above—and its standard culminating question, Would it be morally permissible for you to hit the switch?—has in recent years become a mainstay of research in a subfield of psychology.

And there's the "problem," right there: It's not psychology. It's philosophy.

In November 2016, though, Dries Bostyn, a graduate student in social psychology at the University of Ghent, ran what may have been the first-ever real-life version of a trolley-problem study in the lab. In place of railroad tracks and human victims, he used an electroschock machine and a colony of mice—and the question was no longer hypothetical: Would students press a button to zap a living, breathing mouse, so as to spare five other living, breathing mice from feeling pain?

Right, because our moral calculus involving mice is obviously exactly the same as it would be with fellow humans.

I'm not saying people don't feel sorry for mice. I always feel sorry for the ones that Edgar Allan Purr leaves on the doorstep. But I'd be horrified if he brought us a dead human for a present, instead. Not that he could, but, you know, as long as we're talking hypothetically.

It’s a discomfiting result, and one that seems—at least at first—to throw a boulder into the path of this research. Scientists have been using a set of cheap-and-easy mental probes (Would you hit the railroad switch?) to capture moral judgment. But if the answers to those questions don’t connect to real behavior, then where, exactly, have these trolley problems taken us?

I suppose the answer to that depends on whether you ask a philosopher, a psychologist, a lawyer, or a comedian.

It also seemed a little off that trolley problems were often posed in funny, entertaining ways, while real-life moral dilemmas are unfunny as a rule.

Except that it's the comedian's job to make things funny when they're not.

There's a lot more at the link, of course, but I've banged on long enough. In short, I disagree with the basic premise that it's a psychology issue instead of a philosophy one. Still, as I've noted in here before, it's not completely hypothetical: there are real-life situations where exercising agency can make a difference, one way or the other. So it's worth thinking about.

And it's worth making jokes about, because comedians can often do philosophy better than philosophers.

4 Like • Comment • View 3 comments

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