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:

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

November 2, 2021 at 12:01am

#1020605

Beer Necessities

I don't like being predictable, but this time I just couldn't resist. Besides, if I were always unpredictable, you could predict that I will be unpredictable. Paradox!

PROMPT November 2^nd

You are curator of a museum. This museum has an area of interest to you. Take us on a guided tour of your latest exhibition.

Hello, and welcome to the Museum of Beer.

As you know, beer was one of the earliest and greatest inventions of civilization. While its exact origins are unknown, there is evidence of beer production dating back about seven thousand years.

But you can find more about that in another exhibit. Over the millennia, brewing improved and evolved, and by the 7th century C.E., European monasteries got in on the act. Many of them still produce fine fermented malt beverages, most notably the abbeys of Belgium, and it's that country we're focusing on in this exhibit.

There is a wide variety of Belgian beer styles, not all of them produced by monks; the country has its share of microbreweries and industrial-level beer manufacturers. Now, we all know that AB/InBev, the corporate successor to Anheueser-Busch, producers of such disgusting swill as Budweiser and Bud Light, is based in Belgium. Do not allow this to confuse you. Yes, some of their corporate history includes Belgian breweries, but they do not make Belgian beer.

Okay, yes, they do manufacture Stella Artois, but that hardly counts.

No, when I speak of Belgian beers, I mean the saisons; I mean the Trappist styles like the dubbel, the tripel, the quadrupel; I mean the delicious and hilariously-labeled Delirium Tremens beer produced by Brouwerij Huyghe -- here, you can see the pink elephants

adorning the bottle. I mean the strong golden ales, the blondes, the ever-present darks. And even the sours: the lambics and the Flemish reds.

But one of the most interesting things about Belgian beers, besides the taste of the brew itself, is the artistry involved in crafting specific glassware for each beer. While Germany has its steins and the US and UK have their pint glasses and handled mugs, many premium Belgian beers have dedicated vessels for pouring and consumption. The most common of these is some variation on the tulip glass

, almost like a wine glass but with a flared rim and wider bowl. But other types exist, such as the tall, thermometer-like glass used by Pauwel Kwak

, that has to be served on a wooden rack because it won't stand up by itself. You know. Much like Kwak drinkers ourselves.

Eventually, perhaps even next year, I hope to visit the country myself and experience Belgian beer culture in all its funky glory. Until then, I can but enjoy the occasional import -- and this wing of the Beer Museum.

4 Like • Comment • View 10 comments

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