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.

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.

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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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October 24, 2021 at 12:03am

#1020012

Hardly Working

I'm a bit hungover right now, so even this concept is giving me a headache.

Why hard work alone isn't enough to get ahead
We're constantly taught the recipe for getting ahead is to put our heads down and outwork everyone else. But that's not quite right.

Some adages seem custom-made to keep people in their perceived social underclass. Things like "Money can't buy happiness," "The early bird catches the worm," and "If a job is worth doing, it's worth doing well." But to me the worst offender of all, the one trap they lay to give people the illusion that they might actually escape the clutches of poverty, is "Hard work pays off."

1) It does not, usually and 2) What even is hard work?

Late this summer, UK author Kate Lister had a realisation that resonated. On Twitter, she wrote: “How old were you when you realised your original plan of being really nice, working really hard, & taking on much more than you should in the hope you would be automatically rewarded for this without asking, was totally shit?”

Even earlier than I realized (US spelling) that Twatter is a cesspool. Because Twatter hadn't even been invented yet. Ah, the good old days.

Despite adages and advice that tell people from a young age hard work will get you everywhere, it really won’t, says Jeff Shannon, an executive coach, and author of Hard Work is Not Enough: The Surprising Truth about Being Believable at Work.

"They" tell you that hard work pays off so "they" can profit off your hard work. Consider this: if hard work were all it took to become financially successful, migrant laborers would be multi-millionaires.

Also consider, for example, the work done by Amazon warehouse workers. I'm sure you've all heard the stories: on your feet, hustling around, following algorithms. Indisputably hard work, though I understand some find it fulfilling. (Get it? Fulfillment center? Badumtish.). Is it going to get you promoted to a cushy manager job? No.

And on the other side, as far as I'm concerned, any job that has you sitting on your ass all day cannot be described as "hard work," no matter how diligently you focus on your task(s). It may be rewarding work, it may be your calling, it may drain you emotionally, it may even be difficult, but having done both physically- and mentally-demanding jobs, I can say without doubt that it is not. Hard. Work.

But it’s not enough to take you all the way to the top. “At a certain point you look around and realise, wow, everyone works hard at this level. Expertise and hard work just become the expectation, and will not help you up the ladder.”

And part of that is that it's not a ladder, it's a pyramid. Not necessarily as in "pyramid scheme," but if you have nine colleagues at your lowly mailroom-equivalent level, and one of you is up for a promotion, then nine of you will be stuck in the mailroom. This continues on up the corporate pyramid. And that's not even taking into account that companies don't promote from within like they used to.

To really get ahead, you need to be doing more than just your job. Realisations like Lister’s often come on the heels of watching colleagues with similar (or fewer) abilities soar, while your career stagnates. More often than not, those who rise are the ones willing to politick their way to the top, while you were too busy just working hard to notice you should be working the room.

And while this article started out promisingly, here it veers into "you need to 'work hard' AND get noticed AND also kiss ass."

This flies in the face of societal training that begins as early as primary school, when students are taught that the quiet, hard workers are those most likely to prosper.

They're told that because the corporate world needs people willing and eager to put in the time and effort, and management would prefer if such people didn't raise a fuss. Hence the "quiet" part. Think of the mule with a carrot tied in front of it like the bait on an anglerfish.

In fact, as Shannon notes, hard work alone typically goes unnoticed after a certain point, because everyone around you is working at or about the same level. If you don’t draw attention to yourself in other ways, it’s easy to fade into the background.

And then, as everyone else begins to take notice of this advice, you're once again lost in the crowd, and then you need to find even more ways to (positively) draw attention to yourself, and the escalation continues.

Unsustainably.

In the end, you're back where we started: career advancement is as much luck as anything. You're lucky if your co-workers can't be as self-promoting as you are. You're lucky if management happens to prefer your style over that of your colleagues. You're lucky if you have just the right personality, or know how to fake it. And let's not forget that prejudice, even unconscious prejudice, plays a role. But I'm not wading into that swamp while nursing a hangover.

So, what, am I advocating a lack of "hard work," whatever the fuck that means if you're in, say, engineering design or IT? Well, no, not unless it's going to burn you out. I'm just saying: don't believe the hype. Think of all the people above you in the hierarchy who are partial or complete slack-offs. Did they get where they are by "hard work?" I don't think so. They also didn't get it by being slackers, but it apparently didn't hurt.

Me, I was able to escape that race most of my life, and I am, admittedly, a slacker. But that's as much luck as winning the race is.

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