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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 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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