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!

less than a minute 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.

Adherennium Plotting something

about 2 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.

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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.

May 29, 2024 at 7:25am

#1071843

Ordinary Average Guy

Well, if it's from MIT Press Reader, surely it has the weight of scientific research backing it up, right?

The Time Hack Everyone Should Know
Much like Dorothy discovers at the end of “The Wizard of Oz,” the key to hacking time is a tool we’ve had all along: Choice.

But then you won't get to meet friends, have adventures, and defeat angry old women. Unless you spend your time playing D&D.

I’m in a complicated relationship with my phone.

Oh, dear gods, this is going to be a screen rant, isn't it? FFS. Another thing we're all Doing Wrong.

So much so that I’ve never used the screen time function, choosing to live in denial rather than dealing with the hard truths of our relationship.

Or you could, and I know this is a novel concept here... just not worry about it.

Imagine my horror then, when my 14-year-old son surreptitiously turned it on and started reading off my statistics from the previous week.

It's your own damn fault for a) reproducing and b) not securing your phone.

We all know that time is our most precious resource: It’s the one thing money cannot buy.

Glib, but demonstrably false on two counts. One, the Beatles weren't lying about "money can't buy me love" (though it can buy beer, which is close enough). Two, if you manage to accumulate enough money, invested wisely, to live off the interest... well, then, you can quit working (also known as retirement) and thus use it to buy time.

And with smartphones in everyone’s pocket these days, we’ve never been more able to track how we use every minute of it.

Wrong pronoun. Should have been "they've," not "we've."

By pressing a button or downloading an app, we can track the time we spend exercising, sleeping, and even scrolling through our social media feeds.

That information is not there for your benefit or neurotic attention; it's there so that companies can track trends.

All of this reads like the "TV will rot your brain" rants from a few decades ago, or the "comic books will rot your brain" rants from before then, or the "radio will rot your brain" rants from before then. Hell, I'm willing to bet that as soon as someone invented fire, someone else was like "That stuff makes life too easy. Kids these days, so lazy!"

Take, for example, the American Time Use Survey.

Taking surveys is one way to waste time, I suppose.

The Bureau of Labor Statistics has been collecting data on a variety of time use markers for almost 20 years.

So they can say things like "Clearly, Americans have too much free time. Let's find ways to work them harder!"

According to their 2020 findings, the average American has enough leisure time to fit in lots of healthy and life-enriching activities: 5.5 hours per day to be exact.

Okaaaaay. Here we go.

The "average American" possesses slightly fewer than 2 legs. The "average American" sports fewer than 10 toes, and fewer than 10 fingers. I can't be arsed to find the precise mean, but considering that amputees exist, it's indisputable that the average (the mean) is lower than the most common number (2 in the case of legs, 10 in the case of fingers or toes). Probably around 1.98 legs and 9.5 fingers/toes. Maybe not those exact numbers, so don't quote me on that, but it's very likely some decimal number close to but less than 2 or 10. What matters in that case isn't the mean, but the mode. And, of course, that people who don't conform to the mode are accommodated in public, but that's not relevant to this discussion.

My point is that the "average" here is probably misleading, and meaningless. If two people are working full-time for every one who is not, that 5.5 hours may be low for the latter and high for the former. Plus, even full-time workers usually get weekends and holidays; I'm not sure if that's included in the average, but you might have a good 32 hours of leisure time (not working or sleeping) on the weekends and 0 on weekdays.

I've gone through the calculations before of a typical (not "average") full-time worker, and deduced that the actual number, when removing times for things like sleep, work, getting ready for work, commuting, dealing with household chores, etc., is closer to 1 hour on a work day. And that's assuming one job, which we just can't.

And that's not even getting into the bigger, less pedantic point, which is that as soon as you add "healthy and life-enriching activities" to your day, that time no longer counts as leisure time, because you're doing stuff you feel like you have to do instead of stuff you want to do.

Or that for some of us, screen time is "life-enriching." I've been using some variant of a computer every day since February of 1979. Yes, even before the internet. I played games, sure, but I also learned programming basics and other stuff. And then the internet came along and, well, here I am.

Pant. Pant. Back to the article.

But the survey also showed that we often eschew healthy, happiness-driving activities for passive, screen-based ones.

Newsflash: how we achieve happiness is very personal. I get that some people who use computers all day at work may feel the need to disconnect in downtime (I was never one of those people; I used computers for work and play.) I also get that some of those who do manual labor might want to crack open a beer and watch spurts or whatever. It's different for all of us.

The average American spends 22 minutes a day participating in sports, exercise, and recreation; 32 minutes per day socializing or communicating; and 26 minutes per day relaxing or thinking. In contrast, they spend 211 minutes per day watching TV. That’s 2.6 times more time watching TV than exercising, relaxing, and socializing combined.

Without reiterating my "average" rant from above, I'll just note that this is presented as a Bad Thing and Something That You're Doing Wrong.

Studies have shown that heavy TV watching is related to lower life satisfaction, and that people who use social media the most are also the most socially isolated.

Assuming those studies are even valid, how much of that is because you're watching shows or spurts, versus how the inevitable barrage of ads designed to make us feel worthless and incomplete is making us feel worthless and incomplete?

The article goes on to play more funny games with averages, and ends with some not-so-bad psychology that, while I call it not-so-bad, still won't work for everyone because, just like not everyone has 2 legs, not everyone responds in the same way to the same psychological mind-games.

For instance, the article seems to make a distinction between socializing in person and socializing over the internet. I've been at this for long enough that I don't make that distinction.

In the end, if you feel like you're too focused on something, whether it's exercise or screen time, change it if you want. If you don't, then you didn't really want to; you just thought you wanted to.

Relax. Have a beer (or whatever equivalent you use to say "fuck it" to the world). And don't let other people dictate how you should be spending your time... unless they're paying you.

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