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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 15 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 17 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

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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