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').

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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

February 16, 2023 at 12:01am

#1044915

Nothing at All

Today's article, written in the Before Time (way back in 2018) but still pertinent, is from The Conversation.

In praise of doing nothing

Sounds like my kind of article! Except I don't actually spend my time doing "nothing." I spend it doing "nothing productive," and I like it that way.

In the 1950s, scholars worried that, thanks to technological innovations, Americans wouldn’t know what to do with all of their leisure time.

Whew! Good thing we dodged that bullet!

Yet today, as sociologist Juliet Schor notes, Americans are overworked, putting in more hours than at any time since the Depression and more than in any other in Western society.

Did I say "still pertinent?" I probably should have said "even more relevant."

It’s probably not unrelated to the fact that instant and constant access has become de rigueur, and our devices constantly expose us to a barrage of colliding and clamoring messages: “Urgent,” “Breaking News,” “For immediate release,” “Answer needed ASAP.”

Oh, for fuck's sake. You had me, and then you lost me. Blaming the technology while acting like we don't have agency makes it sound like we have no choice. Well, perhaps on some level we don't, but I'll save the "free will" arguments for later; another article in my queue tackles that.

I've been connected to the internet since the early 90s, more hours on than off, and I know how to turn off notifications. Hell, I even opted out of Amber Alerts.

Over the past decade, I’ve tried to understand the social and psychological effects of our growing interactions with new information and communication technologies, a topic I examine in my book “The Terminal Self: Everyday Life in Hypermodern Times.”

Oh hey, another book promotion. No wonder this article is free.

In an age of incredible advancements that can enhance our human potential and planetary health, why does daily life seem so overwhelming and anxiety-inducing?

Endgame capitalism.

Why aren’t things easier?

Endgame capitalism.

It’s a complex question...

No, it's not. It may have a complex answer (I'm not convinced of that), but it's a pretty damn simple question.

...but one way to explain this irrational state of affairs is something called the force of acceleration.

If you borrow terms from physics, at least try not to mix your metaphors. Oh, well, at least they don't call it "quantum."

Whether it’s in the grocery store or in the airport, procedures are implemented, for better or for worse, with one goal in mind: speed.

Yeah? There's nothing wrong with speed, especially when I'm in line at the grocery store waiting while the lady in front of me is arguing over the price of a goddamned onion.

The problem, in my view, isn't the quest for speed, but the demand for ever-increasing efficiency and, yes, productivity.

Noticeable acceleration began more than two centuries ago, during the Industrial Revolution. But this acceleration has itself … accelerated.

Acceleration, in physics, is the rate of change of velocity—that is, the derivative of velocity. You can also take the derivative of acceleration to get its rate of change. There's a mathematical term for that, too, which I find appropriate in this context: "jerk."

Unchecked acceleration has consequences.

It's not "unchecked." There's always a limit.

At the environmental level, it extracts resources from nature faster than they can replenish themselves and produces waste faster than it can be processed.

All the fossil fuels we've dug up in the last, oh, 200 years or so represent the stored solar energy of billions of years of living entities. I won't venture to guess just when it's all going to run out, but I'm certain it's less than a billion years. Replenishment was never in the cards.

At the personal level, it distorts how we experience time and space.

I mean, lots of things do that. Drugs, e.g.

American workers’ productivity has increased dramatically since 1973. What has also increased sharply during that same period is the pay gap between productivity and pay.

I was promised that, by now, we'd be working two days a week and fishing five. I was also promised a flying car. (Don't bother sending links to flying cars; those are prototypes and I don't have one.)

Instead of maintaining pre-automation levels of productivity, we decided that to compete, we had to work more instead of less. And don't even get me started on how, while wages have stagnated, the excess productivity has enriched the owner class.

Clearly, acceleration demands more work – and to what end? There are only so many hours in a day, and this additional expenditure of energy reduces individuals’ ability to engage in life’s essential activities: family, leisure, community, citizenship, spiritual yearnings and self-development.

I get by perfectly well without most of those things, and I'm not overworked. Family? No thanks. Community? I have the internet. Spiritual yearnings? Don't make me laugh.

And yet, rants aside, the basic premise of the article still appeals to me:

In a hypermodern society propelled by the twin engines of acceleration and excess, doing nothing is equated with waste, laziness, lack of ambition, boredom or “down” time.

Gotta worship at the altar of Holy Productivity.

Like I said, I don't do "nothing." And I don't get bored easily.

As legends go, Isaac Newton grasped the law of gravity sitting under an apple tree. Archimedes discovered the law of buoyancy relaxing in his bathtub, while Albert Einstein was well-known for staring for hours into space in his office.

Sadly, at least two of those are apocryphal. Still, mythology reflects human attitudes, and those clearly show that our attitude used to be "sometimes you just gotta sit."

Danish researchers found that students who disconnected from Facebook for just one week reported notable increases in life satisfaction and positive emotions.

That's funny. I've been disconnected from Facebook since before this article was written. Never even had a Twatter account.

Different social movements are addressing the problem of acceleration. The Slow Food movement, for example, is a grassroots campaign that advocates a form of deceleration by rejecting fast food and factory farming.

Now that's a phrase I haven't heard in a long time. A long time, indeed. I wonder if it's still going on. Not that I care. Factory farming is essential to feeding the teeming eight billion of us, which makes that "movement" the absolute picture of privilege (if it's not astroturf to begin with). And as "slow food" is always immediately followed by "movement," all I can think about is constipation.

French philosopher Albert Camus perhaps put it best when he wrote, “Idleness is fatal only to the mediocre.”

All due respect to Camus, but the vast majority of us are mediocre.

But that doesn't mean we can't improve through idleness.

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