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 15 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 17 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.

April 20, 2020 at 12:24am

#981575

You're FIREd

This article is over a year old, but still relevant to our current situation.

https://www.vice.com/en_us/article/gya8bx/i-tested-the-saving-technique-that-pro...

I Tested the Saving Technique That Promises Retirement at 40
The FIRE—"financial independence, retire early"—movement is all about hoarding enough cash before investing it and living off the dividends.

Not entirely. It's also about not being in debt, and investing along the way.

Like you and every other person in the world, I am no good at New Year puritanism. Two months in, I'm still often drunk and just as unfamiliar with kettlebell squats.

Already I like the author.

Founded in America in the 1990s, the "FIRE" movement stands for "Financial Independence, Retire Early."

A twisted way to force an acronym if there ever was one.

My salary is just under the London average of £34,000 [$44,711]. Taking half my monthly wage for saving, along with rent, bills, and monthly subscriptions like Netflix and Spotify, I'm left with £377 [$495] for the month—a budget of around £12 [$16] a day to cover food, travel, and any other expenses.

So, we're not looking at someone with a great deal of privilege. Saving half your income is easy when you're making 100K a year (quid or bucks, doesn't matter) and don't have kids. Not so much if you're not pulling in even the average for your area.

My first week's budget is almost blown out by the post-midnight beers I buy in the blurry hours between New Year's Eve and New Year's Day.

Yep, definitely wanna have a pint with this guy.

To balance it out, I spend the rest of the week either at work or at home, not spending much, and not really doing much, either. Time stretches out when you have nothing to do.

And this is why it's still relevant today. For people who are lucky enough to still be working, anyway. Most of us aren't supposed to go out, and even if we could, nothing's open.

One aspect of the FIRE movement that appealed to me was its anti-consumerism. A number of the blogs I read argue that if only we bought less and stopped caving into immediate desires for things, we could save more and abandon the daily grind.

Which is easy to say and harder for most of us to actually accomplish.

Some combine the FIRE movement with environmentalism, arguing that if you buy less stuff you can save money and the planet at the same time.

This hypothesis has some merit to it, as I'm sure you've seen in articles about how clean some cities suddenly are.

I'm reminded of an article I saw about Tory MP Dominic Raab eating the same Pret baguette for lunch every working day of his life.

I laughed when I learned enough French to translate the name of the ubiquitous food shop "Pret-a-Manger." I also find it amusing that this conservative British MP is basically being French.

At first, it was easy to fill our weekends by doing free stuff, though a few weeks in they also start to drag. There are only so many free art exhibitions you can wander around before it gets boring. We spend one Saturday getting drunk by downloading some pub and bar apps that give you a free pint just for signing up, while the Dusk app also gives you a free drink a day, but we found the bars it was available in were a bit shitty.

Maybe you need to acknowledge that "doing stuff" can include sitting at home and reading or playing video games? And I hate to admit it, but one of the easiest ways to save money is to cut way back on the drinking. Fortunately, I'm past that stage, but now I don't drink as much as I sometimes imply - mostly because I didn't get in the habit when I was younger and broker.

If saving half of your mediocre salary seems a bit extreme to you, as it did to me, you might be wondering if there is a middle ground between extreme frugality and carefree spending. Both bloggers I spoke to agree that FIRE isn't a binary choice—saving 50 percent of your salary each month isn't the only way to achieve a comfortable future—but you don't need another article on "money hacks" and the benefits of meal prep to know that you should probably be saving a bit more than you already do.

Yes. Life is not binary, as I keep yammering on about. You pick a target level and try to stick to it. Even 10% is good, and there are religions that urge or require a 10% tithe. How much better if you can save that for your own security instead?

With global warming triggering the collapse of entire ecosystems, and the constant barrage of other anxiety-inducing news, now does not seem a particularly prescient time to hinge your future on the continuing ability of the global economy to offer returns on your savings at its historic average—and even if it does, some people take issue with the maths behind FIRE.

Prescient. Also, I know a few people who might wish they'd saved up more before they lost their jobs.

Writing for Bloomberg, Jared Dillian argues that "the biggest issue with the FIRE movement is that it's the ultimate bull market phenomenon. FIRE seems to work because the stock market has gone straight up." He says that even if this continues, using the figures that underpin many of the FIRE movement's assumptions, "it's not going to be any fun living on a shoestring budget and watching your nest egg decline in value by 30 percent to 50 percent."

And behold, the stock market did crash, as it inevitably does. However -- and I respect Bloomberg as a financial source -- investors need to internalize the idea that bear markets are temporary, and if you're investing for the long term, they become barely a blip. I agree that seeing it happen to your savings is no fun, but as long as you're not investing on margin, and if you're not using the money in the next 3-5 years, you'll be okay.

Beyond the issues with the maths behind FIRE, to me it just doesn't seem worth it to offset the present in such a drastic way, to reduce life to a series of dispassionate financial decisions, for a shot at something that might never come. "If I just drop dead of a heart attack in six months time, that would be a bummer because I've deferred my spending for the future," says Whiter. "What if I never get to experience that?"

And that is a sentiment to which I can relate. I'm pretty sure I've said many times something to the effect of "sure, I can do everything right, and then get hit by a bus, and then I'll be a dead Puritan." Each person has to find his or her own balance between want-it-now and wait-for-it.

But I also want to address a larger issue, and for this it doesn't matter if you're living in the US, the UK, or some other industrialized economy, and that's the relationship between consumerism and the economy in general. Just as many of us recycle, or work to find other ways to reduce our carbon footprint, on the grounds that "every little bit helps," our spending is what keeps the economy going. This is the sort of thing where an individual contribution doesn't amount to much, but put a lot of them together and you end up having an effect - possibly even bringing on the bear market the above quote touches on.

Governments have, in fact, discouraged hoarding cash, ostensibly for this reason. But also, at least here in the US, it seems like there's another reason, intended or not: here, health insurance is largely tied to one's job. This gives a massive advantage to employers; they can demand people work in shitty conditions and for lower pay, just for the availability of health insurance. These employers like this situation, and the larger ones (including insurance companies) have influence in politics. So it's reasonable to think that The System is working to keep people working, no matter what. Work, and you get insurance. Spend, and you're helping to keep the economy going. Don't hoard cash; don't save, because you don't want businesses to fold, do you? But then you suddenly lose your job, and then what?

This situation, as is now apparent, is unsustainable. A healthy economy isn't about stock prices; it's about an individual's effort being both good for the individual and good for society. But saving is good for the individual; not so much for the economy. Spending is good for the economy; not so much for the individual.

One touchstone for whether something is ethical or not is: what happens if everyone does it? For example, I chose not to have kids. What if everyone chose that? Well, then, the human race would, perforce, come to a screeching halt (that's not going to be what ends us, though). But turning it around, what if everyone decided to have all the kids they could? Well, that would lead to Malthusian collapse. In that example, the only ethical choice is to let people make their own choices. So, what if everyone decided to -- I'll go ahead and use the acronym -- FIRE?

That, I leave as an exercise for the reader.

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