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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10 Most Recent Blog Comments:

Elycia Lee ☮

about 3 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 16 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 19 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 21 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

April 26, 2020 at 12:19am

#982073

Nightmare Waking

When NatGeo got sucked up by the Murdoch empire, I quit paying attention to it. I fully expected articles to crop up with headlines like, "Climate Change: Myth or Hoax?" and "Crony Capitalism: Boon or Blessing?" And I wasn't far off the mark.

Well, it went to Disney when Disney acquired a bunch of Fox properties, so it's okay again now.

https://www.nationalgeographic.com/science/2020/04/coronavirus-pandemic-is-givin...

The pandemic is giving people vivid, unusual dreams. Here’s why.
Researchers explain why withdrawal from our usual environments—due to social distancing—has left dreamers with a dearth of “inspiration.”

Personally, I haven't noticed any difference. I'm still biphasic, and I still occasionally get sleep paralysis. I suspect exercise helps with the latter, because my exercise options are limited right now and the condition seems worse again. But I know how to live with it. Anyway, my own experience is probably because my life hasn't really changed much; this is not true for most people, and the article I linked above isn't the first time I've seen something about disrupted sleep patterns in this period.

Science has long suggested that dream content and emotions are connected to wellbeing while we’re awake. Bizarre dreams laden with symbolism allow some dreamers to overcome intense memories or everyday psychological stressors within the safety of their subconscious. Nightmares, on the other hand, can be warning signs of anxieties that we might not otherwise perceive in our waking lives.

Some good things come from even the worst of times. In this case, maybe we'll gain a better understanding of sleep and dreams. But I don't think any of the above sentences should be stated with the certainty that they exude.

At least five research teams at institutions across multiple countries are collecting examples such as Weller’s, and one of their findings so far is that pandemic dreams are being colored by stress, isolation, and changes in sleep patterns—a swirl of negative emotions that set them apart from typical dreaming.

I wish someone would approach me about doing a sleep study. "Hey, you like to sleep, right? How'd you like to get paid for it?"

The neurobiological signals and reactions that produce dreams are similar to those triggered by psychedelic drugs, according to McNamara. Psychedelics activate nerve receptors called serotonin 5-HT2A, which then turn off a part of the brain called the dorsal prefrontal cortex. The result is known as “emotional disinhibition,” a state in which emotions flood the consciousness, especially during the rapid eye movement (REM) stage of sleep, when we typically dream.

This connects to yesterday's entry, then.

Multiple studies have shown that our waking activities create a slide reel of memories that influence the content of our dreams. Emotions carried over from the day can influence what we dream about and how we feel about it within the dream itself. Reducing or restricting sources of everyday memories—by being stuck alone in quarantine—may limit the content of dreams or cause the subconscious to reach for deeper memories.

Bold hypothesis, there. Let's see if that plays out.

In Barrett’s latest sample of coronavirus dreams, which she began collecting in March with this survey, some participants reported dreaming they caught the virus or were dying of it. In another set of dreams Barrett collected, participants replaced fear of the virus with a metaphoric element, such as bugs, zombies, natural disasters, shadowy figures, monsters, or mass shooters.

Or as I like to call it, "Thursday."

For those experiencing coronavirus nightmares, there is growing evidence that so-called “dream mastery techniques” can alleviate their suffering.

The article's worth a read if this is affecting you or someone you know. And probably the hyperlinks therein are useful, too, but I couldn't be arsed to check them all.

Last night, I had one of those dreams where you wake up, go about your business, and then realize that you're still dreaming. Sometimes when that happens, I can consciously influence the dream - say, by having Halle Berry show up in the leather catsuit. But this time was different. No matter how I tried, I couldn't make anything happen in the dream.

It was a bit like being awake.

4 Like • Comment • View 1 comment

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