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.

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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.

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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.

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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.

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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').

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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.

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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

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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.

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February 25, 2021 at 12:01am

#1005237

The Art of the Stout

You'd think that, as a long-time beer snob, I'd like Guinness.

The Untold Story of the Guinness Toucan

You'd be wrong.

I mean, it's not bad, and it's certainly an important beer for many reasons. I'm sure I'd like it better if I were to go to Ireland and drink it from the tap there. And given a choice between Guinness and that watered-down rice water that passes for "beer" that's mass-produced in the US, I'll take Guinness every time.

But I made the mistake once of drinking a pint at a concert. The mistake was, before the concert, I'd gone to a taphouse and had a really good stout. By comparison, the Guinness just didn't measure up.

Like I said, though, it's a culturally important beer for many different reasons, one of them being just how long it's been brewed.

Beer drinkers are used to seeing eye-catching beer labels — some good, some not so good. But stroll past a beer fridge and you might see a label that’s simple, nostalgic, and a throwback to the 1930s: the Guinness toucan.

Beer art is flourishing right now. Graphic artists are probably having a field day with all the logos and labels and tap handle designs, to say nothing of posters and taphouse decorations.

“The Guinness family did not want an advertising campaign that equated with beer,” the UK History House writes. “They thought it would be vulgar. They also wanted to stress the brew’s strength and goodness. Somehow it led to animals.”

At least they didn't pick an elephant.

There was the pelican that stole everyone’s beers with the copy, “My goodness — my Guinness!” Also the sea lion that had a habit of stealing Guinness. And the turtle that steals Guinness on its back. A lot of thirsty animals, basically. None earned as much fame as the toucan, however.

I don't think I was ever aware of these other mascots.

The toucan’s brightly colored beak contrasts nicely with two dark glasses of Guinness (almost always two glasses, playing off of the similar sounds of “too can” and “toucan”).

As I'm only aware of the ones with draft pints, I never would have made the "two can" connection. No cans, no pun.

The toucan and its gang of animal friends graced Guinness ads for decades. Then, in 1982, Guinness stopped working with S.H. Benson and dropped the animals. In recent days the toucan has made some appearances, including a limited-edition can released in 2016. But it primarily lives on in the memories of Guinness lovers and collectors.

They're more than happy to sell toucan merchandise in dedicated Guinness stores (yes, these exist), along with the famed bar towels.

So yeah, even with the proliferation of beer choice nowadays, there will always be a place for Guinness. But that's as much a testament to the power of marketing than to the beer itself.

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