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 17 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 9, 2024 at 10:00am

#1070769

Super Stew?

I've done entries about categorization problems before. Is it a sea or a lake? A dwarf planet or a planet? A hill or a mountain? Is a hot dog a sandwich or a taco, or is it just a hot dog? Well, here's one that I've wondered about for a long time, yanked from the website at Country Living.

Stew vs. Soup: Everything You Need to Know
They're both served in a pot—so what's the story?

Served in a pot? Maybe in your country. In my country, they're cooked in a pot and served in bowls.

The article is from January of 2020, but surely nothing's changed since then.

When it's cold outside or you're not feeling well—no matter the time of year—there's nothing more comforting than a hearty soup recipe or a slow cooker stew recipe.

Sure there is: the actual soup or stew, preferably made by someone else because it's cold outside and I'm not feeling well.

On that we can agree—but when you break it down, do you really know the difference between stew and soup?

No, but I guess you're about to tell me. Incidentally, I don't really care; I only care that it tastes good.

The main difference is the amount of liquid that's used for each. In a soup, the ingredients will generally be completely submerged in liquid, while in stews, they're just barely covered.

Okay, but what about when you have ingredients that float? Carrots, e.g., or pasta. You look at it, it's only partly submerged. Like the iceberg, not like the Titanic.

The main component of a soup is the liquid. Soups can be brothy (think: a timeless chicken noodle soup), puréed (classic tomato purée soup), or creamy (potato cheddar soup, anyone?).

I'm not reproducing the links here, but I'm starting to think this "article" is just meant to promote their recipes.

Oh, and to further complicate matters, there are other names for soups. Bisque, a subcategory of creamy soup. Chowder, same. Pho, which I maintain stretches the definition of soup, because it cannot be eaten with a spoon alone unless you have superpowers.

Soups are generally easy and fast to make—some can be made in as little as 20 minutes!

20 minutes? To source all the ingredients, prep them, fumble around with cookware, etc.? Hell, if your batch of soup is voluminous enough, it'll take 20 minutes just to bring it to a boil before you turn it down to a simmer.

On the other hand, I've made soup in 3 minutes: 1) Open a can of Campbell's 2) Mix it with a can full of water 3) Nuke for 2 minutes.

The general rule is that no ingredient should be larger than a soup spoon.

See, that's a tough rule for pedants like me. First, define "soup spoon." After that, consider ramen. Each ramen noodle is, well, a noodle. Even if it were possible to fish out the noodles one by one, bits of them would hang over the sides of the spoon, making it larger than a soup spoon. Sure, if you got it all coiled up, an individual noodle could maybe fit in one spoon, but who does that?

Also see: French onion soup.

Stew tends to be more complex and takes longer to cook than soup.

Betting we could find counterexamples if we tried.

Unlike soups, which rely mostly on water or stock as the main liquid, stews can contain beer, cider, or wine for additional flavor.

I put brandy in my French onion soup.

But. The end of the article addresses a long-standing question, one of great import and whose definitive answer could change the world.

Is chili considered a soup or a stew?

Still, I don't care; I consider it delicious.

4 Like • Comment • View 2 comments

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