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 7 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

about 14 hours ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day 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 ✍️

a day 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

a day ago
In response to "Gray Matters"

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

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
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

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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