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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 10 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 12 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.

June 3, 2020 at 12:44am

#984895

Fun, Fun, Fun

Today, because it looks like we need it (and because it came from my queue at random), we're going to talk about having fun.

https://qz.com/quartzy/1564012/having-fun-is-a-virtue-not-a-guilty-pleasure/

Having fun is a virtue, not a guilty pleasure

The article opens with a helpful picture of someone pouring champagne into a glass, with the caption: Though Judeo-Christian philosophy presents work as divine, there’s nothing inherently sinful in pleasure.

Not to start a religious debate or anything, but I'm convinced that "Judeo-Christian philosophy" is a misnomer for several reasons. You know what the biggest difference between Jews and Christians is? "The Jesus thing." Well, that's a difference, but no. "Christians can eat delicious bacon." Okay, point for Christians. But no. It's this:

Speaking in the most general terms, Christians hold the belief that humans are inherently sinners, and can only be redeemed through Jesus (or Jesus and good works in certain denominations). Jews tend to the belief that humans are inherently good but sometimes do bad things that they can atone for.

It's a huge difference when you think about it.

But the second-biggest difference is that Jews aren't afraid to have a good time.

Okay, I'm probably being unfair to most Christians there. I'm thinking specifically of American Christians, who generally come from a Puritanical background, and Puritans are deathly afraid that someone, somewhere, is having fun.

Again, I'm not trying to start a religious argument here; just giving my impression based on what I've seen. But I wasn't raised Christian, so that work ethic, that idea that "work is divine" that the caption speaks of, well, that was never drilled into me like it was into some of my friends. Work is important, yes, but it's also important to have fun, and in my family, it was learning that was emphasized. Despite being irreligious now, that attitude is still with me.

Also, as far as I know, only the Jews have a holiday where the whole purpose is to get so drunk you can't tell the good guys from the bad guys. Point for Judaism. Doesn't quite make up for the no-bacon thing, though.

And that's all I'll say to refute the idea that there is a monolithic, overarching "Judeo-Christian philosophy," as if the dozens, if not hundreds, of different sects, sub-sects, and individual congregations of each wasn't enough to dismiss the idea.

But I was talking about the article, which is about having a good time.

Imagine you had control of another person’s life, and could dictate their activity every hour of the day. Would you condemn them to spend 80% of their waking hours at a desk, repeating mind-numbingly boring tasks and depriving them of close relationships? Hopefully not—sentencing someone to such a miserable life would be immoral.

A lot of what God is reported to have commanded would be absolutely immoral if it were imposed by a human authority. To me, that's the definition of God. And oh, man, I'm going to start a religious argument with that, aren't I? Okay, look, I'll say this: I'm really not trying to diss anyone's religion or lack thereof; again, I'm just calling it as I see it.

“Some people worship beauty, some worship political identities, and others worship their children,” wrote Derek Thompson in the Atlantic. “But everybody worships something. And workism is among the most potent of the new religions competing for congregants.”

Well, not dissing anyone's religion except maybe that one.

Such widespread workaholism in part reflects the misguided notion that having fun is somehow an indulgence, an act of absconding from proper respectable behavior, rather than embracement of life.

There was a time, however, when I would have been considered a workaholic. It wasn't that I didn't want to do other things; it was that there was so much to do at work and not enough time to do it all. This got especially bad when I was running my own company. But I never thought I was being virtuous by doing it, except in the sense that I wanted to meet my obligations and fulfill my promises.

Indeed, Western society only began to shun pleasures, and to view work as virtuous, thanks to Christian ethics. Whereas the Greek philosopher Aristotle thought that work made you a worse person, Judeo-Christian philosophy holds that work is a means of imitating God’s act of creation, and so views work as divine. Millions of people who’ve abandoned theist practices still hold onto the religious belief that work is inherently virtuous.

I won't belabor the hyphenated adjective anymore, but yes, even those who have become atheists and come from a Christian background can sometimes worship work in a sense, just as I hold dear the act of learning.

Others suggest that having fun is frivolity, and claim to enjoy spending all their time at the office. Of course, work is fun, and a life without meaningful labor can be exceedingly miserable. But that doesn’t imply that work should be the only enjoyable activity in life. Working hard is part of the good life, as employment can bring fulfillment and financial freedom. There comes a time, though, when those financial rewards should be put to use for the sake of leisure.

People talk a good game about work-life "balance," but I don't think they're looking at it right. As a friend of mine once put it, "I don't live to work. I work to live."

Grant notes that Nietzsche, who espoused the value of life-affirming activities, is the most well-known modern philosopher to revive the importance of loving life.

Nietzsche also had a truly epic mustache. I mean, really, look him up and LOOK AT THAT EPIC MUSTACHE.

We are limited individuals, with a narrow sense of self, if we only allow ourselves to pursue the corporate mission of our employer. We need to spend hours devoted to art galleries, or playing cards, or drinking fine wine, to create richly developed selves with an identity unmodulated by company culture. To live well, it’s not enough to simply fall into enjoyment occasionally; we should pursue the joyful activities that make us happy and make life worthwhile. There is nothing inherently sinful in having fun.

Which is great, if you get paid enough to have leisure activities other than sitting and staring off into space.

I don't really have a profound point to make with all of this, except: even in these trying times -- especially in these trying times -- it's important to have a bit of fun and not feel guilty about it. And no, it doesn't have to involve booze.

But it helps.

And now, the results from yesterday. I promised a Merit Badge to whoever answered the question "What the hell are these dodecahedrons, anyway?" in the way that I deemed, in my own opinion, to be the best.

You guys did NOT make that easy. I loved each and every one of the nine comments I got on the subject, even the utter non-sequitur about a sport I know next to nothing about. Though, I have to say, I was just a teensy bit disappointed that no one guessed "prototype Magic 8-Ball."

I mean, seriously, it was hard to pick a clear favorite. Some made me chuckle, like "Gladiator wiffle ball." Some impressed me with their thoughtfulness, like "It's a shape sorting toy." (I once read an article that said that the wheel, commonly accepted as one of the most useful inventions of all time, probably got its start as part of a children's toy, so we shouldn't discount the importance of play toys in ancient times; see the above screed about "fun.") And the "souls of your enemies" one was right whimsical; it would make a good fantasy or horror story.

So what clinched it for me is that I'm a gamer, so the D&D explanation appealed to me (I have to admit when I first saw the dodecahedron, I thought "D&D die," but then again, that's what I think whenever I see any Platonic solid); consequently, Chibithulu (Alyssa) is getting a Merit Badge.

Here's the winning entry from Chibithulu (Alyssa):

They were clearly playing some sort of game with these dice like creations. Maybe it was like marbles where they had to kick the other dice out of the circle, or maybe they had imaginations enough to be playing an extremely limited version of D&D. "What kind of neanderthal do you want to be?" "The smashy kind." "Well, we're all the smashy kind, so let's play!"

This was fun (heh). Again, thanks for all the great answers. I'll probably do it again soon, so everyone will have another chance at an MB.

Until then, have fun

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