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 14 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 21 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day 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

Dec 24, 2024 at 5:16pm
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

May 22, 2024 at 9:26am

#1071517

Jumping Through Loopholes

I'm usually a law-abiding citizen. Or, well, I try to be; sometimes it seems laws are designed so that if They want to get you for something, They can find a reason.

But some laws are fundamentally unjust, and need to be broken. Cracked provides some examples in:

6 Loopholes People Used to Break the Law and Get Drunk
Technically, if you’re on a train, everything is legal

6. Pay to See the Blind Pig

During Prohibition, a bar could not legally operate and sell alcohol. No one could legally sell alcohol (without receiving special exemptions, such as for medicinal use), or manufacture alcohol, or transport alcohol. The law didn’t ban drinking alcohol, however, or handing the stuff to someone else without charging them any money.

It could, of course, be argued that it's hard to drink alcohol if one is prohibited from buying, making, or moving it. But, at least here in the US, lawyers thrive on technicalities.

But suppose an establishment were to hand out a drink for free and charge customers for something else? Say, they charge a customer for some entertainment — for instance, the chance to look at a marvelous animal. As for the drink the barman serves the customer, well, no one’s purchasing that.

Not mentioned: how the pig got blind. Look, I'm all for eating the tasty critters, but mutilating them doesn't fly. This was nearly a hundred years ago, though, and people didn't generally think that way.

This idea is why one alternate name for a speakeasy is a “blind pig.” And if you’re wondering why the police would ever be fooled by this, know that plenty of police didn’t really care about Prohibition laws and were possibly drinking right along with everyone else.

Which may explain why some cops love do do drug busts: it provides them with free product.

5. Instructions on How to Absolutely Not Make Wine

They had this sort of thing for beer, too.

Individual families were still allowed to make a limited amount of wine, but if you were to sell people the raw materials for making wine, along with instructions, you might find yourself in trouble.

This would be like selling fertilizer with instructions on how to ensure it never becomes a bomb. Except there's good reason to keep people from making bombs.

“After dissolving the brick in a gallon of water, do not place the liquid in a jug away in the cupboard for twenty days, because then it would turn to wine.”

These days, of course, the surest way to get a certain group of people to do something is to tell them not to do it.

4. Stick Everyone on a Train

Even when alcohol is legal in the country, you need a license to sell it. One British gin maker, Tapling & Meegan Distilling, dutifully applied for this license, but it was taking too long to get approved. So they did the only reasonable thing and turned to steampunk.

Now, there's a story idea.

Within all the country’s many alcohol regulations is a line of law saying the usual license requirements do not apply to trains in motion.

The downside of this is obvious: motion sickness exacerbated by drunkenness.

3. Let’s Call Beer Something Else

Complex regulations define exactly what beverages are, which is why you generally cannot put lemonade in a bottle and sell it as tomato juice. In Texas, they had a rule about beer: It could not contain more than 4 percent alcohol by volume.

Ah, yes, the old "name a thing something else to get around regulations" trick. No wonder fermented beverage categories can be misleading.

A lager that appears to be beer by most conventional definitions would be labeled, in fine print, “In Texas, malt liquor.”

I'm pretty sure some states still have ABV maximums for beer. As the article notes, Texas isn't one of them. But it's not a good look for a state that prides itself on limiting government interference in peoples' lives. Which, incidentally, they clearly don't do.

2. Turning Nightclubs into Pop-up Restaurants

This next law lasted from 1935 to 2000, in Ireland, a place not entirely unfamiliar with alcohol.

Now that's a cheap shot. Pun intended.

Establishments were not allowed to serve alcohol at night unless they also served a meal.

This sort of thing has been the law here in the US, off and on, depending on where you are. Hell, even my state has a version of the rule, which is why you'll technically never find a bar in Virginia; only restaurants that happen to serve booze.

The dish of choice at these clubs? Curry.

Yeah, that couldn't have ended well.

1. The Inedible Sandwich

That Irish policy hearkens to an older and famous law from New York. Way back in 1896, the same time that they made the controversial decision to raise the drinking age from 16 to 18, the state passed a law saying bars couldn’t serve alcohol on Sundays.

Like I said.

Bars, which served no actual food, qualified for the exemption by offering a sandwich. Not sandwiches, but a singular sandwich that a bar would pass from customer to customer without anyone eating it. This was named the “Raines sandwich,” after John Raines, the senator who wrote the law.

I'd heard about this loophole before, of course. It still amuses me.

The takeaway here is that, if you're clever, you can find a way around unjust laws without flagrantly breaking them. And I approve.

3 Like • Comment

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