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 2 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 4 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.

December 7, 2022 at 12:01am

#1041481

It's Not Even Half Full

And now it's time for Adventures in Itoldyaso.

Want to be happy? Embrace being miserable

Ironically (or whatever), this article is from the Before Time: March of 2019.

It’s good to be positive, sure.

I'm positive I'm gonna need a citation for that.

But that doesn’t mean being sunny side up all the time. Keep your frown right where it is. Let your bitch face rest.

Ever since I first heard the expression "resting bitch face," it's bugged me. For starters, it's sexist (even if it's also applied to dudes). For finishers, it implies that one should always be "on," or not resting, when other people are around, and that's exhausting for some of us.

Americans seem to be down these days.

Oh, if only they'd waited about a year.

Duhigg argues that the source of privileged people’s unhappiness lies in overly high expectations and too little practice struggling with obstacles early on. Thompson, meanwhile, blames devotion to work—the fact that people have replaced God and family with careers and callings as the source of meaning in their lives. Because we no longer want to make “time for happiness,” he says, we are busy, confused, and sad.

Okay, I can follow the logic that one secret to happiness is to set your expectations low enough that you can only be pleasantly surprised. That's a big part of my life philosophy. But that other guy doesn't know what the hell he's talking about.

Struggle helps cultivate resilience, and American “workism” is misguided.

I'm not going to argue either of these points, but lots of people struggle and never get any benefit out of it. As for "workism," however they want to define it, well, some people find their meaning in work; telling them they're misguided is only going to get them defensive. The problem comes in when people are expected to always be working, just to make ends meet.

But Duhigg and Thompson also ignore a more fundamental issue. It’s clamoring for happiness that makes people miserable.

And on that point, I can't disagree. Or I'd be an even bigger hypocrite than I already am.

Brock Bastian, a social psychologist at the University of Melbourne School of Psychological Sciences in Australia, argues in a 2018 paper in Emotion that trying too hard to be happy yields the opposite result.

Ah, yes, another name that, if one of us fiction writers came up with it, we'd be accused of being unrealistic.

“Happiness is a good thing, but setting it up as something to be achieved tends to fail,” Bastian told Time.

Which is kind of what I've been saying. Nice to have an actual scientist looking into it, though. Sure, I'm aware this may be confirmation bias on my part, but I'm still willing to hear alternative viewpoints.

The best relationships involve pain...

Oh, yes, baby, yes.

Oh, they're not talking about S&M. Damn. Well, maybe the point of pain (as the author probably intended it) is to better appreciate the pleasurable parts.

...and the greatest jobs are also tedious.

Eh, citation needed.

Nothing can be fun all the time, and some stuff that ends up enjoyable may seem dreadful while you’re doing it. Pretending otherwise, expecting a steady sense of pleasure and satisfaction, only compounds suffering.

Easy, there, you're veering off toward Buddhism. But yeah. Like, doing your taxes, for instance: massive pain in the ass, and I don't know too many people who actually enjoy the process. But then you get to enjoy not being in jail, so there's that.

Also, things like cooking can be difficult or tedious while doing them, but hopefully the outcome is enjoyable as well.

Although the pursuit of happiness is enshrined in the US Constitution, to act upon that right guarantees displeasure. Desire causes suffering, or so the Buddha believed.

Aaaaaand now your GPS plopped you straight into Lake Buddha.

In fact, this whole section is about that philosophical tradition.

Still, liberating yourself from the expectation of happiness lightens your load. It makes life a little easier when you are realistic but resolved, rather than deluded, desirous, and determined to have the impossible. By calculating discomfort and struggle into the mix, you can remain cautiously optimistic, knowing there’s surely trouble ahead, but that you will face it with grace.

Or, and bear with me on this, maybe you can be, instead of cautiously optimistic, hopefully pessimistic. That is, if you expect the worst possible outcome, then you're either right (which is nice), or pleasantly surprised (which is almost as nice).

Happiness is necessarily not lasting, and if you chase it, the emotion will elude you. It’s precious and momentary. That is what makes it so delightful.

Being transient doesn't make it less real. In fact, I think its ephemeral nature makes it all the more worth savoring. But I do agree that chasing it is counterproductive.

Beyond understanding happiness more deeply, we can train, actively cultivating perspective with practices both ancient and new. If meditating your way to the understanding that emotions are like clouds flitting across the sky of your mind isn’t your thing, for example, there’s a bot that will talk to you about feelings and train you to reframe your thinking.

Isn't the future a wonderful place to live? We can talk about our feelings with artificial intelligences.

The Woebot...

As soon as I read that, I became transcendently happy. Seriously. What a name! I'm only a little bit angry that I didn't come up with it.

...is an app designed by Stanford psychologists, based on Cognitive Behavioral Therapy.

Aaaaaaand there goes the happiness, fleeing like a ninja in the night. CBT my ass.

Over millennia, no one has avoided suffering. It’s unlikely any of us will be the exception. So the wise thing to do is to accept this sad but funny fact. The joke is on all of us and the quicker we are to see the humor in it, the better our chances of sometimes having fun.

Of course humor is essential.

And, for me, so is beer.

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