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:

Wordsmitty ✍️

about 2 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

1 Like

Annette in France

about 13 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

about 17 hours ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

about 17 hours ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

about 22 hours ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

about 23 hours ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

about 23 hours ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

a day ago
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

2 Like

Cubboo!

a day ago
In response to "Get Baked"

Yes! I discovered this a few years back (slitting the potato about 1/4 - 1/2" lengthwise, then one or two slits across the longer slit.) It came out wonderful! It seems like this was called an English jacket potato, or something like that.

2 Like

April 3, 2024 at 9:25am

#1067483

Sumer in the City

Thank Atlas Obscura for this one, which, again, I'm sharing mostly for the hell of it. Because it talks about two of my favorite things. No, not science and math; beer and food.

Ordering Off a 5,000-Year-Old Mesopotamian Menu
A community effort brings ancient flavors to modern Texas.

Texas has forfeited any claim to being "modern," but whatever.

Among the campus buildings at Rice University in Houston, Texas, one curious structure stands out... It’s called a mudhif, and for Iraq’s Marsh Arab ethnic minority, it’s a traditional village meetinghouse where disputes are settled and important gatherings are held.

What Marsh Arabs are doing in Houston is explained in the article.

Alrawi describes the dinner as a way “to complete the setting” of the mudhif by paying homage to the structure’s ancient origins. Rather than modern Iraqi food, the dinner menu is based on recent archaeological discoveries from Lagash. Located in modern-day southern Iraq, Lagash was a major Sumerian city home to some 100,000 people in the third millennium BC.

While that might not seem like a huge city by today's standards, I think it was quite a lot of people for 5,000 years ago. The far better known Babylon, in the same general area, might have reached 200,000, but that was several centuries later.

...the central grain was barley, used to make both bread and another Mesopotamian staple that the team in Houston set their sights on recreating: beer.

And now they're speaking my language.

Though the beer of ancient Mesopotamia bore little resemblance to the delicious magical concoctions of today, it was fermentation of malted barley. The article goes into some of the process.

“Whenever you put an archaeologist together with a chef or a brewer,” says Lao, “they feed on each other.” Recreating the Mesopotamian meal has been a team effort.

Now, that's an image I might have done without.

Also, as the article points out, it's probably impossible to faithfully recreate all of the recipes. Moreover, we're talking about working-class food, the ancient equivalent to street tacos, hot dogs, and the like. Food availability and price keeps changing; a commonly-touted example is that lobster used to be considered trash that only poor people ate, while now it's treated as a luxury item (I don't know how true that is). So do you recreate the food, or the experience? Lots of cultures have come up with their own working-class food, but the commonality there (pun intended, as always) is that it's all based on what's abundant and cheaply available.

At least until it becomes widely known as delicious, at which point it ceases to be working-class food. Like how you might pay $10 for a stadium hot dog, or how I pay $30 for a pizza.

But I feel like it brings history to life if they try to recreate the recipes with the foods then involved. Few people love memorizing the facts, dates, and minutiae of history, but get people together for a meal, and stuff will stick in their memories.

That said, I have little interest in trying original Sumerian beer. It's enough for me to know how the beverage got its start.

2 Like • Comment • View 3 comments

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