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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Brandiwyn🎶

a day ago
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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 8, 2022 at 12:03am

#1038835

Waste Watchers

Stop liking what I don't like!

16 Things That Are a Total Waste of Money

FROM THE LOFTY perch of old age, and after a lifetime of thrift, I declare that I am qualified to comment on how not to waste money.

OK Boomer.

We’ve all heard the reports: Most Americans live paycheck to paycheck, a large number can’t come up with $400 for an emergency, and there’s no money to save for retirement and other goals.

"Most" isn't very helpful. It could mean anything between 51 and 99%. Turns out the current number is about 64%

as of last January. But I don't know what the basis of that statistic is. Does it only include wage-earning employees? Contractors (such as Uber drivers)? Whatever; I'll grant it's a lot of people. Also the link provided later in this article is from 2017 and quotes something like 78%, which seems high.

Most of that data comes from surveys where people are, in effect, saying they don’t have enough income. My curmudgeonly reaction: Stores, fitness centers and entertainment venues are packed with shoppers, many of them buying unnecessary goods and services.

Well? They don't have enough income. Also, how do you know that "stores, fitness centers and entertainment venues are packed with shoppers" unless you go there yourself?

It’s a funny thing: I have yet to see Warren or Bill in one of the many local spas.

Oh, do you live in Omaha or Seattle? Warren Buffet is known to live a relatively modest lifestyle for all his wealth, but I don't know (or care) much about what Gates is up to. I do know that when you're that kind of rich, you can have your own spa installed in your supervillain lair.

Most Americans live like no other people on earth. We have more and bigger stuff: Larger houses, bigger vehicles, more shoes. And, in my not so humble opinion, we can’t tell the difference between needs and wants, between necessities and desires—and we sure can’t defer gratification.

Who's this "we" shit?

As for deferring gratification, in my opinion, if you keep doing that, you'll die unfulfilled.

All this leads me to one conclusion: We’re unable to control our spending or manage our money.

This is by design. Even if you don't take into account the systemic problems with employment right now, advertising, which is ubiquitous, ensures that we're always feeling unsatisfied, no matter how much junk we have. Sure, you can lay the blame on the people on the receiving end of that, but that's like blaming someone for being fat when the only food available to them is ultra-processed high-calorie junk food.

Here are 16 things that this 75-year-old considers big money wasters:

And of course this is what the article's all about. I'll skip some in the interest of not writing a thesis here.

1. Tattoos. They’re an admitted obsession of mine. What will they look like when you’re my age? From what I’ve heard, a good tattoo artist charges $200 an hour.

No young person ever considers what they'll look like at 75. Hell, the way things are going, they won't make it that far anyway. I don't have tattoos either, but I don't tell other people what to do with their skin.

2. Vacations. Hey, everyone needs a break. But you don’t need to go into tuition-level debt to have a good time. Your kids will survive if they never visit the Magic Kingdom.

Disney World is undeniably expensive, but calling it "tuition-level debt" is hyperbole beyond even my usual levels.

4. Restaurants. Eating out, or buying $4 designer coffee, is expensive and—wait for it—it’s also a luxury. Skip that daily $4 coffee and after 30 years you’ll have more than $121,000, assuming a 0.5% monthly return.

What you don't seem to understand is that some people work two or three jobs just to make ends meet, which leaves absolutely no time or energy for grocery shopping or cooking for yourself or even making your own coffee. Also, a 0.5% monthly return (which is 6% a year) is a hell of an assumption; it does track with the long-term average post-inflation gains of the stock market, but there are no low-risk investments that can provide that sort of return.

7. Credit cards. When people say they live paycheck to paycheck, does that include purchases put on credit cards that aren’t paid off that month? In that case, they’re spending more than their paycheck—and what they buy will cost them the purchase price, plus a hefty interest rate.

I gotta agree on this one. But credit cards aren't the problem.

8. Lottery. The lowest-income groups spend the most on lottery tickets, wasting hundreds of dollars a year—about the same as that $400 emergency fund they don’t have.

We've discussed this here before. Yes, the math shows that it's not worthwhile. But life is about more than math; the lottery is the only thing that gives some people hope. Sure, some folks have a gambling problem, but let's address that instead of being like "oooh gambling ooga booga."

10. Shoes. Surveys suggest the average American woman owns more than 25 pairs of shoes, which they admit they don’t need. So why buy so many pairs? It seems shopping and wearing trendy stuff makes us feel good.

"So I want to ensure that you don't feel good."

14. Holidays. Somehow, every December, financial caution goes out the window and we pay for it the following year. But my pet peeve are those inflatable characters on lawns that cost hundreds of dollars. Talk about blowing money.

They annoy me, too, but not in terms of how much money people spend on them. And yet, I understand that they make some people deliriously happy, and who am I to argue with that?

16. Haircuts. The average haircut reportedly costs $28.30 in a barber shop. Many men pay a lot more. Nowadays, nearly a third prefer a “salon.”

Admittedly, I got off that hamster wheel. Long-haired hippie freak for the win.

I kind of agree with some of what he's saying, but here's the issue I have:

Pretty sure I've said this before, but whatever. I once saw a video on YouTube (so it may or may not be true, but that doesn't matter for my point) where a guy talked about spending $22,000 on Superb Owl tickets one year. Now, that's tuition-level spending, unlike Disney World. It's not something I'd want to do. I wouldn't spend $22K on any entertainment, not even Springsteen tickets. But, so what? It's what he wanted. Presumably, it was his dream, his life goal. You know what I would spend $22K on? A beer tour of Europe. Lots of people would think that's a massive waste of money, but it's my dream and one of my life goals, assuming WWIII ends in my lifetime.

The point being that we all have things that we want, over and above needs. Some of them involve spending money, because despite what you're told by people with a vested interest in keeping you poor (companies, churches, etc.), money can buy happiness, at least briefly (all happiness is brief). I won't agree with all your choices; you won't agree with all of mine.

Then there was the time I've also talked about when, driving through Nevada from Reno to Vegas (one of my all-time favorite drives), I made a pit stop in Beattie. I'm in this little shop waiting for the guy to get off the phone so I can pay him for something I'm buying that I don't really need. Looking around, behind the counter, I see a cat carrier. So of course I'm hoping to see a cat. But this little hand comes out and grips one of the carrier door bars.

Guy gets off the phone. I'm like, "Is that a monkey?"

"Yup. Wanna see?" He proceeds to take out the monkey, a young capuchin named Hannah, who then starts climbing all over the counter, then me, then him, all the while making cute little monkey noises. Meanwhile, he's asking me about what I'm doing in Beattie.

"Oh, I'm on the way to Vegas."

"Not to gamble, I hope. Might as well take a lighter to your cash."

As I was, indeed, on my way to Vegas to gamble, I didn't say anything. But I did note, to myself, that he was dissing me for how I spend my entertainment budget while he had a literal monkey on his back. Not to mention happily taking my money for a T-shirt (which I still have and occasionally wear) and some ghost pepper sauce (which was delicious).

Gambling is, for me, an entertainment expense, like the guy who spent half a year's salary on the Superb Owl. Only not even close to that level. The guy in Beattie blew his money on an exotic pet (legal in Nevada from what I understand). It's easy to rag on someone for spending money on things you consider frivolous; it's also easy to justify spending your own money on something other people consider frivolous.

So I ended up writing a dissertation anyway, I guess. But I can't let this go without noting my own list of things that I, somewhat younger than the self-described curmudgeon above, consider big money wasters.

1. Kids. Utter wastes of money.

2. Sports. By which I mean going to live sporting events, paying for cable, subscribing to ESPN, etc.

3. A boat. Or any other leisure vehicle.

4. Art. Sure, it's nice, but that's what museums are for.

5. Cable TV. Never did have it. I'd be paying $200 a month for something I'd rarely use, and still have to suffer through commercials when I did. No thanks.

Or how about some things that I do spend money on that other people would consider frivolous? Well, no, this has gone on long enough already. If you've been following along, you already know most of it.

Point is, though, sure, people could manage their money better. Or employers could stop being so damn stingy. But when budgeting, it's always important to reserve some discretionary funds, if you can, for things that could bring you joy, or at least relief from the crushing boot of endgame capitalism. Even if those things are items or experiences that some old guy considers a waste.

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