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 21 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🎁

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

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

August 8, 2024 at 12:05pm

#1074935

Iraq, A Hard Place

I don't know much about this source, Ulukayin, but I'm going to assume the article is true because I want it to be true. Yeah, I'm too lazy to be skeptical today.

5000-Year-Old Sumerian Pub and Beer Recipe Unearthed

Archaeologists have unearthed a Sumerian pub in Iraq dating back to about 5000 years ago. Among the remains, there is also a tablet with a beer recipe.

As we all know, beer itself is far older than that. It's at least as old as civilization. Hell, some claim it's the reason for civilization, and I'm not going to argue with that.

The discovery was made in Lagash, an ancient city-state in southern Iraq, by a team of archaeologists from the University of Pennsylvania in the USA and the University of Pisa in Italy. Examining the remains, the team reported that they thought the pub was around 5000 years old.

Now my inner skeptic is starting to wake up. "They thought the pub was around 5000 years old?" Did they, you know, do any dating or did they compare architectural styles or what?

It is estimated that beer was also consumed in the pub, although analyzes continue.

So they're not sure if it was actually for beer. Just that a beer recipe was there. I mean, sure, that's pretty solid circumstantial evidence; why have a recipe if you're not going to actually brew beer?

I'm giving them a pass on "analyzes;" English isn't their first language.

Archaeologist Holly Pittman of the University of Pennsylvania said that the most common beverage consumed by the Sumerians was beer. She also stated that a tablet with a recipe for beer was found in a nearby temple.

Okay, now I'm getting confused. Was that a different tablet?

So I guess I kicked my inner skeptic into action, after all. Don't get me wrong; finding the remains of any 5000-year-old building is pretty cool by itself. But did they really have to sensationalize it by making us visualize a public drinking establishment? I don't even know if those existed at the time; I was under the impression that beer was more of a sacred thing, so finding a recipe in the temple makes more sense.

Final note about where this was discovered, the ancient city of Lagash: One of my favorite breweries is Allagash, in Maine. I like them because they produce Belgian beer styles. I don't know where they got their name. But from now on, I'll be thinking of them as Al-Lagash.

5 Like • Comment • View 2 comments

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