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 9 hours 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

a day 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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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

September 12, 2024 at 10:13am

#1076648

Deep Gruits

As I might have mentioned before, beer is as old as civilization. Just like civilization itself, it's gone through some changes, as this article from The Takeout demonstrates.

The Beer That Had Medieval Europeans Hallucinating

It's no secret that drinking too much beer can make your head feel like it's spinning.

What? Damn, I thought that was the room.

But things could get way worse for beer drinkers in the Middle Ages, when the drink could cause full-on hallucinations.

Which isn't always a bad thing. Hell, sometimes, that's the goal.

The culprit for such false visions, sounds, and sensations was an ingredient called black henbane, which brewers often added to ales.

You don't want to try black henbane? What are you, chicken?

At the time, the general practice was to make beer without hops.

I've noted before that hops, now considered one of the essential ingredients of beer (though some people take it too far), are a relatively late addition to the art of brewing. By "relatively late addition," I mean it's still older than my country, but it still took something like 11,000 years to figure out.

Instead, beer makers flavored their goods with plants, herbs, and various other flora, collectively known as gruit.

In case you're wondering (I did, when I first saw it), you pronounce gruit almost exactly like fruit. Which makes me want to do a Guardians of the Galaxy parody with "I am Gruit."

Perhaps the most dangerous characteristic of black henbane is that it causes dry mouth, causing the beer drinker to be increasingly thirsty, and thus, wanting to drink more beer.

Shit, don't say that out loud! If they figure this out, AB/InBev would absolutely bring back black henbane as a way to sell more pisswatery beer-like liquid and thus make more profits.

Which reminds me of a thing my dad told me long ago. In New Orleans, during the Depression, he said there was a place that served big plates of Cajun spiced shrimp for free. The catch was, beer was a dime, and if you've ever had Cajun-spiced shrimp (or anything), you might understand how this was a remarkably effective business model for the 1930s.

This was problematic, because while a small bit of the plant could cause a drunk sensation, too much of it could result in extreme hallucination, and even death.

Ah, yes, another case of "the dose makes the poison." Lots of things are harmless or even beneficial in small quantities, but large ones will kill. Vitamin A comes to mind.

By the beginning of the 16th century, the presence of black henbane in beers and ales began to dwindle. Around this time, brewers were discovering that it was cheaper to make their goods with hops rather than gruit, and that the resulting beer had a longer shelf life (even though most beer doesn't last as long as you think).

The origin of this perfect marriage of hops and malt is still a bit of a mystery to me, but their preservative qualities were almost certainly part of the reason hops became ubiquitous. The IPA style was, originally, overhopped so the beer could make it all the way from England to India without becoming too skunked. Why didn't they just make the beer in India? Well, I'm not entirely sure, but this was before air conditioning, and beer needs time to cool after being brewed, and India is mostly hotter than hell.

By the year 1516, Germany had outlawed henbane in the country's beer production, as part of the German Purity Laws, but not every country across Europe followed suit so quickly.

So Germany marched into other countries to impose their purity laws.

Okay, okay, just kidding. They didn't do that. Cheap shot at Germany (and it wasn't even technically Germany at the time). I have German ancestry, so don't give me shit about it.

Here, I'll make up for it. I'm telling this joke now to get it out of my system because after next week, I'll need to refrain lest I get my ass kicked in France: Why are the streets of Paris lined with trees? Because Germans like to march in the shade.

But I digress. The reason beer is defined as it is now (water, malt, hops, yeast) goes back to the Reinheitsgebot,

which some say is the oldest still-enforced food regulation in the world, and is probably one of the reasons why German beer is world-famous.

And if you're still curious about the effects of black henbane in beer, just remember that the average life expectancy for males in the Middle Ages was 31.3 years.

This is where I lost faith in the article. Unfortunately, it's close to the end of it. "Life expectancy" and "average" can be slippery concepts, and, taken together, get skewed by a much higher infant mortality rate back then than any developed country (even the US) experiences today. So that's misleading.

I can believe the stuff about henbane, though. It tracks with what I already knew.

5 Like • Comment

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