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:

tj-Merry Mischief Maker

about 14 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 17 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 20 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

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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