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 2, 2021 at 12:02am

#1022679

How Low Can You Go?

Well, look at that. Two touchy subjects in as many days.

Before I get started, though, a brief rant:

I just had to delete no fewer than 10 emails with a subject line beginning: "Tis the season..." This has got to stop. Yes, I let some businesses send me email because I actually, you know, do business with them. This might change if they can't come up with a more creative way to try to suck money out of people than overused, trite, cliché and lazy lines from annoying earworm Christmas carols.

So it looks like I'm going to have to set up a filter like I did with Mother's Day: Anything with that subject goes directly to trash, unread and unnoticed.

Sick of this shit.

Anyway...

How Low Can America’s Birth Rate Go Before It’s A Problem?

I describe it as "touchy" because, often, population issues can be racist dogwhistles. As in "OMG not enough white people are having kids." I'm just going to state up front here that this is not how I'm using it, as if that's something I'd have to state.

The U.S. fertility rate hit a record low in 2020 — just as it did in 2019, and 2018. Although the COVID-19 pandemic seems to have accelerated this decline, the drop has been underway for years.

I gotta say, I was bracing myself for simpering "lockdown baby boom" stories, but as usual with my cynicism, I'm just glad I was wrong about that.

When the fertility rate falls below replacement level, the population grows older and shrinks, which can slow economic growth and strain government budgets. Today’s babies are tomorrow’s workers and taxpayers: They’ll not only staff the hospitals and nursing homes we’ll use in old age but also sustain the economy by funding our pensions when we retire, paying the taxes that finance Social Security, Medicare, and many other government programs we’ll rely on, and buying the homes and stocks we invested in to build our savings.

Maybe stop thinking about it as a fertility problem and start retooling the economy so it's not a fucking Ponzi scheme?

In fact, low fertility poses some advantages: easing ecological pressures, preventing overcrowding and reducing the infrastructure costs that come with a growing population.

Our economy isn't currently built to allow for a decrease in population. It's set up so that there must always be more and more people. And yet, this is unsustainable; the population cannot grow infinitely. Nor, as the article goes on to point out, is a continuous decrease in population sustainable in the long run. Averaging out to a steady state, that is, one where births and deaths are roughly equal, would be ideal... though we'd argue endlessly over where that level should be.

Personally, I'd put it at maybe half of the current 8 billion (the article is mainly about US population, but I tend to take a more global perspective). There were fewer humans than that on Earth when I was born. But others will disagree with the number. So be it.

Every time I mention that the population could be lower, though (mostly because every single environmental problem we face would be at least partly improved if there were fewer people), I get, "So why don't you start by killing yourself?"

I wish I could understand the mindset behind that. But I don't. I'm not talking about removing people who are already here; I'm talking about not bringing more people into an overcrowded, overheated world. I mean, I understand that someone would want me gone, but for that reason? I simply don't get it.

Later in the article, they talk about some ways to reduce the economic impact from having a smaller population. Then:

Taken together, these findings suggest that we could ease the problems of a low-fertility society if we’re willing to invest in children’s education and better support women in the workforce.

Welp, we're boned, in that case.

Others point out that the problems of low fertility may get thornier when the overall size of the population begins to shrink. “What happens to mortgages in a country where real estate depreciates like a used car because the population is falling and we need fewer and fewer houses all the time? We’re totally unprepared for that,” said Lyman Stone, a demographer and research fellow at the Institute for Family Studies, a conservative-leaning think tank.

Oh no! The price of housing may decline! More people might actually be able to afford to have a place to live! Homelessness might decrease! The horror!!!

And countries like the U.S., Canada and Australia rely on net immigration as well — and could probably continue doing so for decades if they choose to embrace it.

Here's the thing about immigration (and now I'm shifting focus back to the US alone):

When a kid is born, they are completely useless for about 20 years, sucking up resources while contributing nothing. Oh, sure, to the parents they're precious (sometimes), though it's not like the old days when you could put 'em to work on the farm. But from society's perspective, you invest in them for 20 years in the hope that they'll eventually become "productive." And sometimes they don't.

An adult immigrant, however, can hit the ground running and contribute to the economy right away.

To me the choice is clear: immigration is preferable from an economic standpoint. The only way it wouldn't be is if you're concerned about race or some nebulously-defined "culture" (In America? Come on) that's going to change over time anyway.

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