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:

tj-Merry Mischief Maker

about 10 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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?

July 21, 2024 at 8:39am

#1074214

Revisited: "Smiley"

My weekly trip into the archives only took me back two years, this time, to July of 2022, almost to the day. It's a fairly long set of musings on a fairly long article from aeon: "Smiley"

And it is, as one might guess, all about smiles.

Now, it's only been two years, as compared to the many years of history discussed by the article. One of the reasons I do these things is to see how things have changed, and, well, not much has changed in two years. There's some discussion of face masks, which were more prevalent in public in 2022, but that's about it.

I will, therefore, just point out a couple of things I either missed or left out in a failed attempt at brevity back then.

First, as academics are wont to do, the article includes a foray into Latin:

Ancient Romans showed another variant. If we take their vocabulary at face value, they did not distinguish between a smile and a laugh, contenting themselves with a single Latin verb – ridere – for both. Only towards the end of the Roman Empire did a diminutive – subridere – enter the language. This came with the derived noun sub-risus (later, surrisus) a ‘sub-laugh’– a little or low laugh – associated with mockery. It retained this lesser status and this diminutive form, distinguishing it from the laugh as it entered the Romance languages in the High Middle Ages. Around 1300, for example, French contained words for laughing (rire) and laughter (le rire or le ris) and smile (sourire, from sous-rire).

I'll point out now that the Latin ridere is the known origin of our words "ridicule" and "ridiculous," which today don't conjure up much in terms of smiley laughter, but rather lip-raising scorn, such as I occasionally heap upon stupid articles in here. Not the one under discussion today, but others.

Second, it's been bandied about that forcing a smile also forces the underlying emotions of happiness, contentment, or relaxation, and supposedly makes one appear more friendly and approachable.

Neither of these things is always good, in my opinion. Sometimes it's best to work with the emotions you're actually feeling (or so I've heard, at least for actual humans). And one does not always want to look friendly and approachable, because someone might approach you and try to get friendly.

It could even be a symptom of the foul practice of toxic positivity.

And to me, forcing a smile is the exact opposite of relaxation. I'm not suggesting frowning as an alternative, but just... letting it be. Unless you're in the kind of situation where people shouldn't know what your actual emotions are, lest you get intrusive questions like "Why the long face?" or "Okay, what are you grinning about?" Or if, you know, you're at a poker table.

I still pity the retail workers who have to go all day with a fake smile plastered upon their face. Even worse are the ones who are genuinely happy to have to present their customer-service face to the world. I mean... why? Smiley faces are for people who aren't paying attention.

So anyway, the original entry and article are there at the link, and I still think the article's interesting. But I don't think I smiled even once.

3 Like • Comment • View 1 comment

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