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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QPdoll

about 11 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 14 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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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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