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').

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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

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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

September 12, 2022 at 12:02am

#1037631

Sleeper Agent

Today's article is all about my favorite pastime. Okay, second favorite pastime.

Can You Catch Up on Lost Sleep?
You've given up your fair share of sleep—will you ever feel rested again?

Now, the article is from 2008. I'd like to say that doesn't make a difference, but science has a way of correcting itself and changing course. It's not nutrition science, though, so the answer probably hasn't gone from yes to maybe to no to possibly to no to yes during the last 14 years.

Let's do some sleep math.

And you just lost half your readers.

You lost two hours of sleep every night last week because of a big project due on Friday. On Saturday and Sunday, you slept in, getting four extra hours.

Duh, five times two is ten minus four is *counts on fingers* *shrugs*

...still, I was under the impression that sleep deficit doesn't actually work that way. It's not like paying for stuff with a credit card. It's always possible that I was misinformed, however.

Sleep debt is the difference between the amount of sleep you should be getting and the amount you actually get.

If it were a debt you could pay off, I'd still be asleep from the day I retired.

"People accumulate sleep debt surreptitiously," says psychiatrist William C. Dement...

Snicker. I'm sorry. I know we shouldn't make fun of peoples' names. But come ON.

Studies show that such short-term sleep deprivation leads to a foggy brain, worsened vision, impaired driving, and trouble remembering.

Hm. I know something else that causes those symptoms, and it's a lot more fun than merely not-sleeping: booze. That would be my favorite pastime, as I noted above.

A 2005 survey by the National Sleep Foundation reports that, on average, Americans sleep 6.9 hours per night—6.8 hours during the week and 7.4 hours on the weekends. Generally, experts recommend eight hours of sleep per night, although some people may require only six hours of sleep while others need ten.

Seven hours? LUXURY.

The good news is that, like all debt, with some work, sleep debt can be repaid—though it won't happen in one extended snooze marathon. Tacking on an extra hour or two of sleep a night is the way to catch up.

Great. Now: should that be at the beginning or the end, or both?

As you erase sleep debt, your body will come to rest at a sleep pattern that is specifically right for you. Sleep researchers believe that genes—although the precise ones have yet to be discovered—determine our individual sleeping patterns.

Yeah... one thing I've learned in the last 14 years since this article came out is that's not really how genes work. There's not one gene, or a group of them, for a particular trait; rather, it's a complicated mess that's way above my pay grade. Still, I can accept that it's hereditary as opposed to learned.

As I've mentioned before, I seem to be naturally biphasic. I sleep in the late afternoon/early evening, wake up, and sleep again in the early morning. Actual times vary. One of the perks of being retired. The only consistency is that I'm almost always awake at noon and midnight, which is one reason I choose to do these entries around midnight. I say "usually" because alcohol can throw the schedule off, and so does traveling.

That more than likely means you can't train yourself to be a "short sleeper"—and you're fooling yourself if you think you've done it. A 2003 study in the journal Sleep found that the more tired we get, the less tired we feel.

I vaguely recall this being a "life hack" a while back: people insisting that you can train yourself to operate on four or two or whatever hours of sleep. I recall, less vaguely, going "bullshit." As I do with most "life hacks."

Also, apparently there's a journal called Sleep. This amuses me because I imagine the articles are a cure for insomnia.

So earn back that lost sleep—and follow the dictates of your innate sleep needs. You’ll feel better. "When you put away sleep debt, you become superhuman," says Stanford's Dement, talking about the improved mental and physical capabilities that come with being well rested.

Unfortunately, we live in a society that doesn't value sleep, seeing it instead as a necessary evil. When you're sleeping, you're not being productive, and there's nothing more sacred than Holy Productivity. We're conditioned to shame people for sleeping too long, or at the "wrong" times. Naps are scorned. "Sleep is for the weak." "I'll sleep when I'm dead." Fuck all that noise.

And I've done the math in here before, but essentially, considering all the things we need to or should do in a day, most of us just don't have the time to sleep. Theoretically, it's 8-8-8: 8 hours each of work, leisure, and sleep. But almost no one devotes only 8 hours to work; there's a significant chunk of time where you're doing work-related things, such as getting dressed or commuting, that cuts into either leisure or sleep. And if you have kids? Well, forget it. People use lack of sleep as some sort of social cred, too, and that's perverse.

What needs to change is our attitudes about sleep. And maybe shorter work days.

Not that it affects me, not anymore. But I'm incensed on everyone else's behalf.

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