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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June 8, 2023 at 7:55am

#1050787

Staten Island? Yes, It Is.

This one may not be too relevant unless you live in NYC. Or visit there a lot. Or are interested in transportation engineering and its associated politics. I meet two of these criteria, so here's an opinion piece from the New York Daily News.

The subway tunnel from the past to the future: 100 years ago today we started digging a connection between Brooklyn and Staten Island that should be finished

For the uninitiated, NYC has five boroughs: Manhattan, an island (with a very small exclave on the mainland—interesting story there); Queens and Brooklyn (part of an island); the Bronx (mainland); and Staten Island (take a wild guess).

SI is kind of an outlier there; it's more residential and easier to access from New Jersey than from the rest of NYC.

It might help to take a look at a subway map,

(found on Wikipedia), though remember, this is not to scale or up to date. That blue route on the shamefully miniaturized Staten Island is not directly connected to the rest of the system.

The very elegant invitation before me, with the seal of New York City, is for 4/14/23 and reads “Breaking of ground for the Brooklyn Shaft” [of the Staten Island to Brooklyn subway]. But it’s not today; it was 100 years ago, April 14, 1923!

For reference, the famed subway system in NYC opened in 1904, so they were considering connections to Staten Island within 20 years of its initiation.

It's also not entirely a "subway," as much of it runs at or above grade. That 1904 date refers to actual underground subway.

Obviously, this 1.5 mile subway extension of the now R train never got built — although about 150 feet of the aborted tunnel lies under Brooklyn’s Owl’s Head Park. It was hailed as a brilliant idea then and frankly it remains a brilliant idea today.

Reminder: opinion piece.

Not only does Staten Island, a borough of a half-million people, not have a subway, to no surprise it has the highest share of vehicle trips of any borough, matching that of many suburbs.

Half a million sounds like a lot of people (and it is by some standards), but the total population of NYC is about 8.5 million.

As for vehicle trips, because of its lower density, parking is sometimes actually possible in SI, unlike the rest of the city. There is an extensive bus network in the borough, though.

Also, some people consider it a suburb, while Jersey City is jokingly referred to as an honorary borough.

If we turn back the clock to the dawn of the last century, civic leaders envisioned a unified metropolis of the five boroughs that merged into one city in 1898. In 1900, the president of the Citizens Association of Richmond, David Tysen, lamented that transit from the island hadn’t improved in a half century — a familiar plaint to Staten Islanders of today.

Explanation: Richmond is the county; Staten Island is the borough and the island's name. Don't ask me what's up with all the naming confusion; I don't live there.

Since 1900, there have been some improvements, like the Verrazzano-Narrows Bridge connecting SI with Brooklyn... for vehicular traffic.

In 1923, groundbreakings were held in Brooklyn and Staten Island. But, bickering by civic leaders as to whether the tunnel would be for freight and/or people delayed the project through the Depression and then World War II. By 1945 the plan was dead. Instead, four bridges were built linking the island with New Jersey and Brooklyn, but for motor vehicles only.

With all the traffic delays that entails.

If you think this is a “tunnel too far,” remember every other borough has multiple subway lines, some going 10 or more miles deep into the outer parts of the boroughs.

By "deep," the author means "away from Manhattan," which is, of course, the center of the universe. Not that the tunnels are 10 miles deep. Just to clarify.

Considering how much the Second Ave. subway cost for a mile and a half tunnel, nearly $5 billion, I’d say roughly $3 to $5 billion would be enough.

I'd take that with a NYC-sized grain of salt. The 2Av doesn't run under a major shipping channel. Underwater tunnel construction is its own ball game.

Discussion of interborough transit would be incomplete without a mention of that mainstay of NYC culture, the Staten Island Ferry. Build a subway, reduce the need for the Ferry.

In 1993, nearly two-thirds of island residents voted to secede from New York. If city and state leaders continue to ignore “the forgotten borough,” we better get used to saying the four boroughs of New York.

Don't threaten me with a good time. And you're still going to have transportation problems because you're an island. Hey, maybe New Jersey would take you into its loving arms.

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