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').

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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.

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

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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.

2 Like

September 9, 2022 at 12:01am

#1037524

Houses of the Holy

I'm going to talk about a British thing today. No, not that British thing. Or that one. Or even that one. And it doesn't even involve York, but New York.

Physical Graffiti Building
Manhattan, New York
The East Village tenement from the cover of Led Zeppelin’s 1975 album has a tea shop on the ground floor.

Yes, I've been all over Manhattan, but it's a big place and I don't think I ever passed by this building. Hell, I owned a copy of the album as a kid. (Me: "Most expensive album cover ever made." Dad: "And you bought it?") I guess I just assumed the building was in the UK somewhere, because, you know, Led Zeppelin.

Incidentally, expensive to produce, not to buy. And I was just going by some random DJ's assertion; I never bothered to verify it, because this was before the internet and also I was already lazy. Further, for all I know, that was bullshit, or maybe some later album exceeded it. Anyway, as I recall, the consumer price was about the same as any other double LP collection. Pink Floyd's The Wall, e.g., which also had amazing cover art.

Led Zeppelin’s 1975 album Physical Graffiti is considered one of their best, featuring such fan-favorite songs as “Kashmir” and “Trampled Under Foot.” Its release saw a delay due to the complexity of its cover design, which proved difficult to manufacture.

Hence the supposed cost.

Designed by Peter Corriston, the iconic die-cut artwork portrays a symmetrical brownstone tenement block, something that could only hail from New York City, with each letter of the album’s title written on its window.

I always thought the cover was clever. It's one of those things that you can only appreciate as an LP cover. Not nearly as impactful on CD, and certainly not as digitized artwork on the internet.

The building in question is a real tenement in Manhattan, still standing at 96 and 98 St. Mark’s Place. Since similar-looking buildings are not uncommon in the city, more than a few tourists have mistaken other brownstones for the one on the album cover. But once you know the actual location, it’s not hard to find.

Other fun places to visit in Lower Manhattan include the Ghostbusters firehouse

, clear on the other side of the island; and, very close to the PG tenements, The Museum of the American Gangster.

Today, the East Village building is home to a basement tea shop aptly named Physical GraffiTea, boasting a great selection of organic, fair-trade tea and a reputation for offering a wide selection of loose leaf tea and medicinal blends.

A tea shop with a punny name? Now I definitely have to go.

As for my copy of the album, well, it—along with a couple hundred other LPs—was destroyed in a flood, when I was renting a shitty basement apartment not far from where I live right now. This was around the time that CDs were supplanting LPs and cassettes, 1987 or 88 or so. I lost a great deal that fateful night, but the one that still hurts when I think about it, 35 years later, is my album collection.

I never did replace Physical Graffiti. But another thing I'll always remember about it is my old dad (he wasn't much older than I am right now) coming into my room while I was listening to it and going, "That's good music." Considering he would have turned 105 tomorrow, and he grew up pre-rock, I called that high praise.

Can't let a music entry go without including some.

Oh, let the sun beat down upon my face
And stars fill my dream
I'm a traveler of both time and space
To be where I have been
To sit with elders of the gentle race
This world has seldom seen
They talk of days for which they sit and wait
All will be revealed

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