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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Elycia Lee ☮

about 5 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 21 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 23 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

March 17, 2020 at 12:01am

#978319

The Pursuit of Grumpiness

In these days of plague, economic disaster, and impending environmental collapse, it's important to remember to maintain a negative attitude.

No, really.

https://www.bbc.com/future/article/20160809-why-it-pays-to-be-grumpy-and-bad-tem...

Why it pays to be grumpy and bad-tempered
Being bad-tempered and pessimistic helps you to earn more, live longer and enjoy a healthier marriage. It’s almost enough to put a smile on the dourest of faces.

Oh, believe me, this makes me happy. See? This. This is my happy face. (By way of illustration, I look something like this

right now.)

The pressure to be positive has never been greater. Cultural forces have whipped up a frenzied pursuit of happiness, spawning billion-dollar book sales, a cottage industry in self-help and plastering inspirational quotes all over the internet.

All of which have the opposite effect than intended on me.

Now you can hire a happiness expert, undertake training in ‘mindfulness’, or seek inner satisfaction via an app. The US army currently trains its soldiers – over a million people – in positive psychology and optimism is taught in UK schools. Meanwhile the ‘happiness index’ has become an indicator of national wellbeing to rival GDP.

Hire... a happiness expert? No wonder society needed a reset.

The truth is, pondering the worst has some clear advantages. Cranks may be superior negotiators, more discerning decision-makers and cut their risk of having a heart attack. Cynics can expect more stable marriages, higher earnings and longer lives – though, of course, they’ll anticipate the opposite.

As I've always been saying: Expect the worst, and then you can only be pleasantly surprised. Either you're right, which makes you feel good; or you're wrong, which also makes you feel good because the worst didn't happen after all.

Good moods on the other hand come with substantial risks – sapping your drive, dimming attention to detail and making you simultaneously gullible and selfish. Positivity is also known to encourage binge drinking, overeating and unsafe sex.

Now, see? If I were a jovial, positive, upbeat person, I'd binge drink and overeat even more than I do now! On the other hand, maybe I'd have sex.

The article goes on to describe the actual science behind this, which I'm going to choose not to be skeptical about (even though it takes a detour into evolutionary psychology speculation) because dammit, I want affirmation every once in a while. But, you know, feel free to grump about it yourselves.

All these physiological changes are extremely helpful – as long as you get a chance to vent your anger by wrestling a lion or screaming at co-workers. Sure, you might alienate a few people, but afterwards your blood pressure should go back to normal. Avoiding grumpiness has more serious consequences.

Alienating a few people? Sounds like heaven.

In fact, scientists are increasingly recognising that grumpiness may be beneficial to the full range of social skills – improving language skills, memory and making us more persuasive.

See? If I weren't grumpy, I'd have no social skills at all.

Then there's more science describing some more studies to explain just how happiness is a Bad Thing.

So the next time someone tells you to “cheer up” – why not tell them how you’re improving your sense of fairness, reducing unemployment and saving the world economy? You’ll be having the last laugh – even if it is a world-weary, cynical snort.

Now sod off. I'm social-distancing.

3 Like • Comment • View 2 comments

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