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🎶

about 22 hours 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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November 7, 2024 at 10:08am

#1079616

Past and Future Intents

I've lined Aeon articles in here before. This seems to be an affiliated site, Psyche. Apparently, someone loves Ancient Greek.

How to do mental time travel
Feeling overwhelmed by the present moment? Find a connection to the longer view and a wiser perspective on what matters

Bullshit sense... tingling!

But let's give this a chance.

You have a remarkable talent – the ability to step outside the present, and imagine the past and future in your mind’s eye.

Hey, you know what else you can imagine? Things that never happened, places that don't exist, and impossible scenarios. It's a rather important ability for fiction writers. Or city planners.

Some people apparently don't have the capability for visualization, while others can experience it more vividly. But I think it's fair to say most people have some ability to imagine.

Known as ‘mental time-travel’, some psychologists propose it’s a trait that allowed our species to thrive.

Well, at least this is the polar opposite of the "staying in the present moment" crap that's been circulating. I'll give it that.

If I ask you to imagine what you did yesterday, or what you’re planning for tomorrow, you can conjure up rich scenes in the theatre of your mind.

Well. Some of us can. I doubt the author intended to be ableist, but this is a bit like saying, "If I ask you to walk a mile, you can get up and walk a mile," ignoring or forgetting that paraplegics exist.

In the accelerating, information-rich, target-driven culture of the early 21st century, the present often dominates thoughts and priorities instead.

And some people seem to want to make "the present" the only priority.

We need to be present-minded sometimes. However, too much focus on the ‘now’ can also lead to the kind of harmful short-termism that infuses business, politics and media – a near-term perspective that worsens many of the long-term challenges we face this century, such as the climate crisis.

I can't really disagree with that. I've said similar things. Whether we can visualize things or not, we can learn from the past and make plans for the future, and both of those things are important.

But it’s also compounded by a host of unhelpful human habits and biases too, such as our ‘present bias’, whereby we tend to prioritise short-term rewards over long-term benefits (the classic example is the marshmallow test, in which some children can’t resist eating a single treat now, rejecting the chance to chomp two later on).

Yeah... that might not be the best example

to quote.

A longer view provides a deeper, richer awareness of how we fit into the human story – and the planet’s – and reveals just how fortunate you are to be here, right now. The geologist Marcia Bjornerud calls this perspective ‘timefulness’.

I'm not sure I like that name any more than I like its apparent inspiration, mindfulness.

Why a geologist gets to weigh in at all should be obvious: they generally cultivate a real sense of deep time, working as they do with rocks that sometimes predate eukaryotic life.

In this Guide, I’ll share practical tips and exercises that can help you escape the unwanted, short-termist distractions of the present, and discover the upsides of a longer time perspective.

The author proceeds to do just that, and at length. I don't think I need to copy anything else; if you're interested, go to the link (hopefully it won't rot anytime soon). I will note, however, that he does turn back to "mindfulness" at one point in there.

As for my bullshit sense, well, jury's still out for me. I tend to distrust pop psychology (for example, the marshmallow study, above), though that doesn't mean it's all bullshit. But right now, I have plans for the rest of the day because, no, I don't live in the present.

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