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:

Write_Mikey_Write!

a day 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

a day 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.

July 9, 2019 at 1:01am

#962343

Vanguard

Respond to a current news story with your opinion. Share a link to the story if possible.

Opinions are like colons: almost everyone has one, and most of them are full of shit.

That doesn't stop me from having an opinion, of course.

The temptation exists to find some article about some violation of human rights somewhere and comment on it, but really, what's the point? "Ooh, I think violating human rights is bad; look at me!" or "Those people deserve it. See? I'm an asshole."

Or, you know, I could do what I usually do here when there's not a blog challenge running, and link to some scientific discovery or a sober analysis of drinking. But since that's what I usually do, I feel like that doesn't count. Besides, what would my opinion be? "Science is cool," "beer is cool," "the science of beer is cool."

I know! I'll find some political news and display my partisan attitudes, so we can turn the comment section into a mini-Facebook. Not.

So I'll just pick something at semi-random off the front page of fark.com.

https://www.marketwatch.com/story/this-couple-dreamed-about-frugally-traveling-a...

First of all... really, Market Watch? I want to hear about company news and macroeconomic issues from you. Personal finance articles are okay, too, but this is fluff. While we're at it, I know you have a style guide that requires that every time you mention a company you also have to put up its stock ticker and current daily gain / loss, but it's as out of place here as a Corvette at a campground. Facebook's stock price is utterly irrelevant to this article.

Now that we've gotten that out of the way, I'll get to the actual meat of the article. Well, fake meat. The Not-A-Burger of financial news.

Traveling the country, driving past cornfields and mom-and-pop shops until you reach the Grand Canyon. Pulling up the trunk, kicking off your shoes and cracking open a cold one, while dangling your feet into the dusty summer air. Sleeping under twinkling lights. Seems like a picture-perfect life, maybe even one built for Instagram, right?

No. No, it does not. I mean, sure, the traveling part. The dusty summer air, the sleeping outdoors? Nope. Give me a nice air-conditioned (or heated) hotel room at the end of a long day of driving. (Incidentally, they don't list the stock ticker for Instagram because it's not a company; it's part of Facebook.)

It’s known as #vanlife — that is, traveling, or even living full time, in a van — and it’s seductive. At least it was for Lisa, a 43-year-old mother of one who works in pharmaceuticals, and her husband, a filmmaker.

Anything with a hashtag is immediately suspect. Also, those professions? Look, I'm not here to rag on the career choices of the people involved, but the way this is reported is lame. "I work in pharmaceuticals" could mean anything from "CEO of Purdue Pharma" to "crack whore." (Of the two, "crack whore" is more honest.) "Filmmaker" is like "writer" in that it tells you nothing about what the person actually does - that could mean "a guy with a camera" all the way up to "major Hollywood producer."

In this article, I'm getting the impression both of them are somewhere in between these extremes, but it's not exactly clear.

Before they got married in their 30s, Lisa, who lives in Lancaster, Pa., became more interested than she ever had been before in personal finance. She started to follow money blogs and read about icons like Suze Orman. She even started her own blog, called Mad Money Monster, to chronicle her life as she sought to get fiscally fit.

Nothing wrong with any of that. Personal finance isn't something I hear about being taught in schools, and it's a basic life skill, regardless of how much money you make. Or fail to make.

The groups showed “how beautiful your life is going to be if you live in a van, you work remotely, you wake up to a sunset and coffee in your hand,” Lisa, who doesn’t use her last name online to protect her anonymity, said.

Well, I suppose if you sleep all day and wake up to a sunset, your career options are somewhat limited. Personal choice, of course. Hell, I've done it. But it kind of negates the whole "seeing the scenery" part of traveling if you're driving at night and sleeping all day.

The #vanlife movement has a lot of overlap with the “FIRE” movement, which stands for “financial independence, retire early.” At the core of both: a belief that material things don’t make you happy. Ditching your earthly belongings and consolidating, maybe down to living in a van full-time, is where real meaning is, and work doesn’t have to mean sitting at a desk 24/7.

In order for something with a hashtag to be meaningful at all, you at least need the earthly belonging known as a "mobile phone" or similar device. Not to mention the van. By the way, anyone who wants to ditch their earthly belongings, I take donations. You'll live a more real-meaningful life, and I'll have more stuff. Win/win.

Also, anyone who works by sitting at a desk 24/7 is going to die in 3 weeks, and lose their minds way before that. Which might explain some things about corporate America.

Lisa wasn’t naive. She grew up living in a trailer, but seeing others live and travel in Airstreams still seemed exciting, she said.

Those two sentences contradict each other.

They started to nail down the details. They wouldn’t travel full-time; with their daughter and full-time work schedules, that wouldn’t be possible. But they could at least live that #vanlife on weekends and on vacations.

So... you kept your house... and your jobs... and traveled around in a van. Okay. Whatever floats your tits. I like to travel, too; I get it. But I don't think that demonstrates clarity of the whole "ditch your belongings and live full-time in a mobile tiny house" concept.

So they sold one of their cars and bought a 1993 Chevy van, for $3,700, and nicknamed it “Van Halen.”

I want someone to do a survey of what people name their vans. I'm willing to bet that "Van Halen" is, if not #1, at least in the top 5, along with "Van Damme" (or "Damn" as seen in the Netflix NFLX, -1.15% series Sense8), "Van Morrison," and some others I'm too lazy to come up with right now. It's like naming your pug "Pugsley." How about some originality?

Lisa started to see posts in her #vanlife Facebook groups about solar energy, and how to charge phones “when we’re in the middle of nowhere,” which seemed concerning, she said. Others mentioned needing mechanical tools and prowess, in the event that a van breaks down, and she and her husband lacked that.

Did you... did you really not think of those things before putting money down on a used van? At least know how to change a tire and spring for a AAA membership, which is about $10 a month (however useless that is if you don't know you can charge your phone off the van's electrical system so you end up stuck on the side of US 50 in Nevada where there's no cell towers anyway). (I might or might not have some personal experience with that sort of thing.)

Even worse, “We started to see people post things about safety, and that you need to have mace or a gun, and if you have a gun you can’t carry [it] across state lines,” she said. “That started to really spook me, especially traveling with a young girl, our little daughter.”

That's... not entirely the case. Yes, if you're traveling with a gun, you need to be aware of different state laws. If you're traveling with a gun and a child, I expect to read about you on fark.com at some point.

Then she started thinking about the time of year. “I like to be comfortable,” she said. “If we’re going to stay in this van, how am I going to sleep when it’s June? I don’t care if we’re in Lancaster — no matter where you are, it’s hot.”

Again... now? After plunking down the money on the van and conversions? Were you not aware of this thing called "weather" and how it might change over time and in different places?

Suddenly, she found herself looking into the cost of campgrounds. With electricity and water hookups, some of them were almost as costly as hotels, creeping up to $50 or $60 a night, she said. Plus, they’d have to pay for gas to put into Van Halen along the way.

You... didn't... figure... on... gas? At this point, all sympathy I might have once had for Lisa and Mr. Lisa has evaporated. The kid, now - her I feel sorry for. It's tough to start out in life with idiot parents.

She threw in the towel. The family sold the van and kissed their initial plan goodbye.

There's this concept called the "sunk cost fallacy." Look it up. Okay, so they're not total idiots, but close enough.

“We’re so glad we tried it,” she said. “We’re not the typical van-life people. We have a daughter, we have a house. Maybe if you’re a 20-something who doesn’t have those ties, it’s going to be different.”

Hey, Market Watch, you forgot the hashtag. Shame on you. Being in a quote is no excuse; she seems like the kind of person who would have said "hashtag-vanlife."

Her advice for those dreaming about the #vanlife: [There it is. -Ed.] Try it out first. “I would say rent something, or borrow a friend’s, or something like that,” she said. “At least do a weekend overnighter.”

You know, there seems to be a reluctance to take advice from people who have failed at something. For instance, taking relationship advice from, I don't know... me. There's probably a named fallacy for that, too; I don't know. But the fact is, we learn from our failures just as much - maybe more - than we learn from our successes. I mean, every time I see a fluff piece about someone who's lived to over 100 years, they always ask them what their secret is. (Usually, it's booze, cigarettes and promiscuity.) But there's never any asking someone who's about to die at, say, 27, what they'd have done differently. Just once, I'd like to see that, and have them answer something like "I probably shouldn't have had Dave hold my beer and watch this."

So there it is, folks: good advice from someone who's been there.

Hey, I'm not saying "don't sell all your stuff and go live in a van." It's important to follow your dreams.

It's also important to lay off the booze long enough to a) drive and b) plan accordingly.

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