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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Write_Mikey_Write!

about 6 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

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Adherennium Plotting something

about 8 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

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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.

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.

July 13, 2022 at 12:04am

#1035132

Just Once

Speaking of numbers (the musical definition this time)...

A Stanford Psychologist Says He’s Cracked the Code of One-Hit Wonders
What separates Blind Melon from Shania Twain?

"What separates Blind Melon from Shania Twain?" The US-Canadian border.

In September 1992, the band Blind Melon released their self-titled debut album. The record was mostly ignored until a music video for the song “No Rain,” featuring a girl in glasses dressed as a bumblebee, went berserk on MTV. The song rocketed up the Billboard Hot 100 charts.

You know, I'm obviously not opposed to the concept of music videos. Some of them can be quite clever. When that was all MTV did, it was pretty awesome—though I could only watch it at other peoples' houses because I always refused to get cable.

What I have a problem with is a song becoming popular because of the video. It should stand on its own. To be fair, "No Rain" did. Literally. But one thing I've learned is that other people don't appreciate music so much as they love dancing, spectacle, and a performer's brand.

I blame MTV for killing Cyndi Lauper's career, for example. Amazing musician. Really unconventional looks. What they used to call a "face for radio."

Two decades later, Rolling Stone named “No Rain” one of the biggest one-hit wonders of all time.

It never even occurred to me that the band didn't have other hits. I wasn't that into it. But at least I'd heard the song, as opposed to...

Soon after Blind Melon topped the charts, another artist had a breakout moment. Shania Twain released her second album, The Woman in Me, which included the No. 1 hit “Any Man of Mine.”

It's not that I dislike Shania Twain. I just never listened to her music. Wasn't on my radar; still isn't. It's not like her fellow Canadian, Celine Dion, whose music I actively despise.

Whatever the polar opposite of a one-hit wonder is, that’s what Shania Twain turned out to be. She became one of the most consistent hitmakers of her era, and the only female artist ever with three straight albums certified Diamond, meaning more than 10 million copies sold.

Gotta admit, that's impressive. But when it comes to country-adjacent music, give me Brandi

Carlile any day.

For decades, psychologists have puzzled over the ingredients of creative popularity by studying music, because the medium offers literally millions of data points.

Snort. Be attractive and know how to dance. That's it. Musical talent is underappreciated. Hell, if you can put on a spectacle, these days, they'll even autotune your crappy voice.

Is the thing that separates one-hit wonders from consistent hitmakers luck, or talent, or some complex combination of factors? I did my best to summarize their work in my book, Hit Makers.

Oh hey, another ad for a book. Well, that's okay.

This month, the Stanford psychologist Justin Berg published a new paper on the topic and argued that the secret to creative success just happens to hinge on the difference between “No Rain” and Shania Twain.

"This month" apparently being this past April, based on the date of the article.

He used an algorithm developed by the company EchoNest to measure the songs’ sonic features, including key, tempo, and danceability.

Ugggghhhh. "Danceablity" is bullshit.

This allowed him to quantify how similar a given hit is to the contemporary popular-music landscape (which he calls “novelty”), and the musical diversity of an artist’s body of work (“variety”).

Probably using applied mathematics.

Blind Melon’s “No Rain” rated extremely low on novelty in Berg’s research. Dreamy, guitar-driven soft rock wasn’t exactly innovative in 1992.

It hasn't been innovative since the Beatles, but at least it's got the potential for showcasing talent.

By contrast, Twain’s breakout hit rated high on novelty in Berg’s research.

I'm betting "novelty" has to be assessed based on historical context. Hoping, anyway.

Berg’s research also found that musical variety (as opposed to novelty) was useful for artists before they broke out. But down the line, variety wasn’t very useful, possibly because audience expectations are set by initial hits. “After the first hit, the research showed that it was good for artists to focus on what I call relatedness, or similarity of music,” he said. Nobody wants Bruce Springsteen to make a rap album.

Okay, if you're going to bring up Springsteen...

Bruce's first album, Greetings from Asbury Park, had variety in spades. Slow, fast, upbeat, depressing... it's all there, and all with his signature rhyming poetic lyrical style. Far as I can recall, it produced no hits... but Manfred Mann climbed the charts with their overproduced cover of "Blinded by the Light."

Bruce's second album, The Wild, The Innocent, and the E Street Shuffle, was an artistic miracle and a commercial flop.

His third album was Born to Run, and thus history was made.

As for "nobody" wants him to make a rap album, I wouldn't be so sure about that. Last I checked, he was the most prolific songwriter in the business, and has done just about every other genre in existence. I'd buy it.

This second finding about the benefits of early variety is similar to a model of creativity known as explore-exploit. The Northwestern University economist Dashun Wang has found that artists and scientists tend to have “hot streaks,” or tight clusters of highly successful work.

In the case of Einstein, this was absolutely true. 1905 was all Al, baby.

Berg’s and Wang’s research suggests three rules of thumb that may come in handy for creative work.

This is, of course, how the article is relevant here, where most of us are doing creative work of some sort. It's the last paragraph in the article, though, and to avoid lawyers, I'll suggest you go ahead and click through to find these "rules" if you're interested.

Instead, I'll leave you today with what I consider to be the greatest one-hit wonder of all time—from a band that went on to record some truly great, but completely obscure, songs. As a bonus, the lead singer, Paul Roberts, is also an accomplished graphic artist, and as far as I know, he painted all his own album covers.

Pick up your feet
You've got to move to the trick of the beat
There is no elite
Just take your place in the driver's seat

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