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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Write_Mikey_Write!

about 13 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 15 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').

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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.

April 4, 2022 at 12:03am

#1030030

Sleep Like a Baby

I still think the phrase means something different from what people think it means.

How to Sleep Like a Baby as an Adult: Steal the Bedtime Routine of a Toddler
Want the deep, restful sleep of a child? Then put yourself to bed like a toddler, this pediatrician urges adults.

I guess it's referring to how kids can actually wake up refreshed without needing caffeine like normal people. But to me, "sleep like a baby" means to wake up screaming in the middle of the night, thus also awakening anyone else within a five-mile radius.

But okay, let's take it to mean what's in the subhead: deep, restful sleep. Since this is from Inc., I can only assume that they want you to get a good night's sleep so you can be more productive for your boss.

Always-on work culture, pinging phones, Netflix's 'watch next episode' button, and general life stress all made getting a good night's sleep difficult before the pandemic.

There are cures to all of that: Less stressful jobs. Phones with the volume turned off. Even *gasp* giving Netflix a rest once in a while.

Still, even I, possessed of a relatively stress-free life, sometimes deal with insomnia. Not as bad as when I had to make my boss richer, but sometimes sleep eludes me.

We are nearing year two of the coronavirus, but the experts say many people's sleep is still disturbed.

I'm not an expert, but when you combine the uncertainty of the virus, a change in daily routine, a disruption in normal leisure activities, and having to deal with the assholes who don't agree with you on what we should do about it, it's no surprise to me that a lot of people developed or worsened insomnia -- even if you no longer had a daily commute.

Lots of these breathing exercises and calming routines are useful. By all means give them a try and see if they work for you.

Give them a try, that is, with an open mind. If you go in to the practice with my usual pessimism, it'll be a self-fulfilling prophecy of "this isn't going to work."

But, according to one doctor, you don't need fancy techniques or the latest sleep app to achieve deep, restful sleep. You just need to steal the bedtime routine of a toddler.

"One doctor?" One doctor has the cure? Where's the science?

You can probably skip the hot milk and teddy bears, but Merali claims that research shows the essential pillars of a good toddler bedtime work just as well when you're 43 as when you're 3.

Substitute room-temperature whiskey for the hot milk, for starters.

Okay, look. You know I'm a proponent of an alcohol-positive lifestyle. But the above is intended as a joke. Alcohol isn't actually conducive to good sleep. Small amounts disrupt the quality of the sleep you get. And if you consume enough so that you pass out, you will absolutely be out, but it's not as good as actual sleep, and you'll still be tired the next day on top of being hung over. I'd never say "don't do that." But part of living an alcohol-positive lifestyle is knowing the consequences. If you're suffering insomnia, booze is probably not the best choice for dealing with it.

As for weed, I don't know if it helps with sleep or not.

1. Set a fixed bedtime. "There is good quality data that shows regular bedtimes help improve toddler sleep. This is equally important for adults," insists Merali.

I mean... if your schedule is pretty regular anyway, it can't hurt to try this.

2. Take a hot bath (or shower). "A comprehensive study published in 2019 examined 13 different adult studies and found that scheduling a hot bath or shower one or two hours before a planned bedtime significantly shortened the time it took to go to sleep," Merali reports.

I can see how this might help. I've never been a night-shower person. A morning shower was part of my waking-up routine as an owl forced into a lark world. So I associate showering with beginning the day, not ending it. And of course I could never be arsed to shower twice a day. Besides, wet hair means wet pillow (I've heard some people don't get their hair wet in the shower; I'm not one of those people).

3. Apply lotion. This isn't to keep your skin soft (though that might be a nice bonus). "A study that randomized 76 infants to receive a bedtime massage with lotion, a massage without lotion, or no massage, it was the infants that had a lotion massage that experienced longer periods of sleep," Merali says.

...or else it gets the hose again.

Seriously, though, does this person not have to deal with wet and lotion-sopped sheets and pillowcases?

4. Engage in a quiet activity. Bill Gates always reads before bed. Science suggests he's not just expanding his mind but also helping ensure a restful night's sleep.

I'm willing to bet that anyone reading this doesn't have a problem with reading. But I'd really like to see some science on this. If you read before bed, do you not then start to associate reading with sleeping? And then when you try to read a book during, say, a Sunday afternoon, your Pavlovian response is to go right to sleep?

I'm not saying it's right or wrong; I'm saying I don't know.

We know that, for kids, a good night's sleep demands a thoughtful transition from the hubbub of the day.

Still, kids don't have the responsibilities that most of us adults do. They may have nightmares, but not about things like "what if I made a mistake on my taxes?" or "How do I know John from Accounting isn't trying to backstab me?" So there's probably some adult-related stress in addition to the normal ravages of aging playing into why some of us can't sleep well. But from what I can see, these things couldn't hurt to try.

Just don't blame me if you follow this advice and then wake up screaming in the middle of the night, awakening everyone within a five-mile radius. Like a baby.

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