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

September 4, 2022 at 12:15am

#1037286

Cut Me Some Slack

Today's article is a few years old now, but I just found it. It is from the Before Time, and since I'm not sure how relevant it still is, well...

The age of the Influencer has peaked. It’s time for the slacker to rise again

Meh. We'd rather stay in bed. Rising is for hustlers.

It’s hard to remember a time when scrolling through Instagram was anything but a thoroughly exhausting experience.

Yeah, it really is hard to remember. Oh, wait, that's because I've never been to Instagram.

Where once the social network was basically lunch and sunsets, it’s now a parade of strategically-crafted life updates, career achievements, and public vows to spend less time online (usually made by people who earn money from social media)—all framed with the carefully selected language of a press release.

You know, one of the great things about this blog, for me anyway, is that I'm not making any money at it. I'm not trying to make money at it. I'm not selling anything; while I am promoting my point of view, my livelihood doesn't depend on convincing you of anything. I'm not constrained by commercial requirements, only by the self-imposed 18+ rating and my own conscience.

I have no problem with people trying to make money, per se. Just when it's hidden behind a few proxies; that is, if you're pretending not to try to make money.

Well, I despise most ads, too. So there's that.

Everyone is striving, so very hard.

No. Everyone you see is striving, so very hard. The whole thing about slacking is... if we're doing it right, you're not seeing us.

Back in the 1990s, our heroes were slackers: the dudes and the clerks, the stick-it-to-the-man, stay-true-to-yourself burnouts we saw in Ferris Bueller’s Day Off, and Slacker, and Reality Bites.

I admit I haven't seen all those movies, but yeah, The Dude abides.

Fun side story here: A friend of mine has a 12 year old son. He had the kid watch The Big Lebowski to show him the dangers of being a slacker. They watched the movie together, and at the end, the kid was like "I wanna be The Dude."

But somewhere in the early 2000s, the slacker of popular culture lost ground to the striver.

Noughties. Goddammit, the first decade of this century is the Noughties.

I am not immune to this thoroughly aspirational mindset, and you probably aren’t either.

Do I want more money? Yes. Am I willing to be someone I'm not in order to get it? No.

Whether we have side hustles, personal brands, gig economy jobs, or entrepreneurial leanings (I’ve had all four), to survive in the modern economy is to aspire to something much greater than what we are.

I'm not sure exactly what the author intends by "personal brands" there. Is the meaning that of "public persona" or "stuff you're trying to sell?" If it's the former, sure, I have a personal brand, mostly displayed right here on this blog: the alcohol-positive lifestyle, the low tolerance for bullshit, the thinking about science and philosophy, and some things that some people might consider humor.

But all of that is just Me. I mean, aspects of my personality, perhaps magnified or exaggerated, but I'm not trying to be anything I'm not or, as I said above, trying to sell anything.

The internet influencer is the apotheosis of all this striving, this modern set of values taken to its grotesque extreme: Nothing is sacred, art has been replaced by “content,” and everything is for sale. This is true even when the message is swathed in the language of counter-culture: Eco-conscious influencers see no issue in flying long-haul on free trips from brands. Yoga gurus who traffic in anti-consumerist spirituality promote tea brands owned by Unilever.

I've taken to calling them "influenzas."

But as anyone who has lived a few decades knows, youth culture swings like a pendulum. The buttoned-up post-World War II period gave way to the countercultural Free Love generation (arguably the original slackers, as they were the first to have middle class comfort to rebel against). Similarly, the 1980s excess of Gordon Gecko’s Wall Street set the stage for the slackers amid the economic recession of the 1990s, with their flannel shirts, skater culture, Beastie Boys and Nirvana records.

Point of clarification here: There was one (1) economic recession in the 1990s, early on, while Bush was president (not that I'm blaming or absolving him, just setting the stage). This was essentially a product of the eighties. Decades don't always start or end neatly on years ending in 0 or 1. The point I'm making here is that the culture they're describing coincided not with the recession, but with the subsequent economic powerhouse that was fueled by the rise of the internet and other advances in technology.

The 1990 recession was also really mild, as these things go.

But there’s always something to glean from the dominant youth culture of an era. What was cool—what the kids were into—tells us something fundamental about what we valued.

You know why people care about what the kids are doing? Because advertisers want them to. That's it. That's all. (That's also the source of generational labels.) They market mostly to young people because when you get to be my age, you start getting all cynical about advertising. No, kids aren't "cool." They're awkward and trying to figure out what to make of their world, and their place in it. So no, I reject this assertion out of hand.

And, in a modern aspirational marketplace so saturated that fake influencers are now posting advertising-like content that nobody even paid them for, there are signs that our individualist culture of achievement and brand alignment has jumped the shark.

I find this statement hilarious because the term "jumped the shark" came from a sitcom made in the seventies about fifties culture, featuring a character developed as the Avatar of Cool.

To be fair, Fonzie was absolutely the Avatar of Cool. But that's not the point.

If the cycle of history is any guide, once our culture of striving flames out, it may well be time for the slacker to rise again.

And here we have the rare example of a prognostication coming true. Great Resignation? Quiet-Quitting (which should actually be called "doing your job")?

For the internet influencer, everything from their morning sun salutation to their coffee enema (really) is a potential money-making opportunity. Forget paying your dues, or working your way up—in fact, forget jobs. Work is life, and getting paid to live your best life is the ultimate aspiration.

Except are they really living their best life, or just faking it for the clicks?

The phrase “famous for being famous” used to be a derisive slur for socialites; now it’s an entire category of global commerce that has landed the world’s youngest “self made” billionaire on the cover of Forbes: Kylie Jenner.

"Self-made," my fat white pimply ass.

The likes of Kathy Griffin, Ben Stiller, Janeane Garofalo, and Andy Dick typified the ethos of the slacker era.

Also, let's be clear, here: "slacker culture" was also framed, packaged, and sold to us.

What’s more, we’ve never been more skeptical of the very social media platforms that have quietly and powerfully shaped influencer culture.

Yeah, right. Bookface, maybe, but people keep trying to get me to join the Chinese Communist Party. Er, I mean, join TikTok. Same thing.

But as we settle into our roomier trousers and perhaps take a toke of a legally sanctioned weed pen, there’s a less comfortable question to ask: Is it even possible to truly be a slacker anymore?

It's not nearly as slackerish if it's legal.

As Storr writes of our culture’s failed promise: “It wants us to buy the fiction that the self is open, free, nothing but pure, bright possibility … This seduces us into accepting the cultural lie that says we can do anything we set our minds to … This false idea is of immense value to our neoliberal economy.”

I don't have much else more to say, but as far as I'm concerned, that right there is the money quote.

I just think it's buried too deep in the article for anyone with a twenties-era attention span to ever see.

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