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.

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

October 4, 2023 at 10:26am

#1056735

Emotional Breakdown

I know I've talked about this sort of thing before, but this is a new-to-me article.

Struggling with positive thinking? Research shows grumpy moods can actually be useful

At last, validation! I'd celebrate, but then I wouldn't be grumpy anymore.

This is known as “positive psychology” and has recently expanded to accommodate not only psychologists, but also social workers, life coaches and new age therapists.

One of these things is not like the others...

But there is evidence to suggest the approach has a negative side.

Every silver lining has a cloud.

Perhaps the most common advice made by positive psychologists is that we should seize the day and live in the moment.

RAAAAAGE.

Doing so helps us be more positive and avoid three of the most infamous emotional states, which I call the RAW emotions: regret, anger and worry.

Or, and I'm no psych-talker here, but maybe we have those emotions for good reason?

Sometimes I think the "professionals" try to get people to be happy all the time, not for their benefit, but to keep from having to be around negative people. No one (not even me) wants to deal with someone who's negative all the damn time. It's inconvenient for us; why can't they just be happy?

But human psychology is evolutionarily hardwired to live in the past and the future.

I've grown to dislike the "hardwired" metaphor. But remembering the past and being able to plan for the future, well, that's part of what makes us human. If you "live in the moment" (which you can't because there's no such thing as "the present moment"), you're missing a big part of what it means to be human.

Regret, for example, which can make us suffer by reflecting on the past, is an indispensable mental mechanism for learning from one’s own mistakes to avoid repeating them.

I think the problem is when you experience regret and don't use it to improve.

Worries about the future are likewise essential to motivate us to do something that is somewhat unpleasant today but can create gain or spare us a greater loss in the future. If we didn’t worry about the future at all, we may not even bother with acquiring an education, take responsibility for our health or store food.

Likewise, worry is counterproductive when you don't do anything about it. Knowing when there's something you can do, and doing it, is essential. Knowing when to let go because there's nothing you can do, well, that's also essential.

Like regret and worries, anger is an instrumental emotion, which my co-authors and I have shown in several research papers.

No, I didn't follow the link to those papers. But as above, I think there's a difference between useless anger and useful anger.

What’s more, research has shown negative moods in general can be quite useful – making us less gullible and more sceptical. Studies have estimated that a whopping 80% of people in the west in fact have an optimism bias, which means we learn more from positive experiences than from negative ones.

If true, that also enrages me. I didn't check their work, though.

For example, optimism bias is linked to overconfidence – believing we are generally better than others at most things, from driving to grammar.

I'm an inveterate pessimist, but I am better than others at driving and grammar. Worse at a lot of other things, though.

Defensive pessimism, on the other hand, can help anxious individuals, in particular, prepare by setting a reasonably low bar instead of panicking, making it easier to overcome obstacles calmly.

This time, I did follow the link,

which, quelle surprise, turned out to be an older Conversation article about the benefits of keeping expectations low. Which is already part of my psychology.

The next section in the article, Capitalist Interests, well, I can't do it justice by cherry-picking quotes. Suffice it to say that I was already thinking along these lines.

Here's one excerpt:

After all, if we have full control of our happiness, how can we blame unemployment, inequality or poverty for our misery? But the truth is that we don’t have full control over our happiness, and societal structures can often create adversity, poverty, stress and unfairness – things that shape how we feel. To believe that you can just think yourself better by focusing on positive emotions when you’re in financial danger or have gone through major trauma is at the very least naive.

I wouldn't be so generous as to call it naive. More like manipulated.

But here's the "I've been saying this all along" moment:

And then comes the question of whether happiness is really the most important value in life.

I've been saying this all along.

In short, striving for happiness is like fucking for virginity. There's a lot of shit wrong with the world, and in our own lives, and the idea that we should smile our way through it enrages me.

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