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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Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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').

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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.

3 Like

December 28, 2018 at 12:34am

#948342

Leave

Today's linked article turns feel-good stories into feel-bad stories.

https://thinkprogress.org/do-the-impossible-never-complain-live-the-dream-the-da...

Stories like this keep popping up on Twitter like zits on a prepubescent forehead: The sunshiney announcement about the GoFundMe for the guy with leukemia who can’t pay for his own medical costs. (He is employed by an organization whose owner has a net worth of $5.2 billion.) The dad who works three jobs to support his family saving up to buy his 14-year-old daughter a dress for an eighth grade dance. The college student who ran 20 miles to work after his car broke down and whose boss rewarded him for this effort by giving him his own car.

Do you get a sinking feeling when you read these stories? This feeling like, while of course you are impressed by the tenacity and generosity on display, you still want to vomit?

Behold, the rise of the feel-good feel-bad story.

I don't disagree with the thrust of the article. Providing for others may be more rewarding to accomplish directly, rather than indirectly through mandates. But it's been obvious to me for a while that our system is broken.

And yet - and here, I'm going to venture into controversy - I'm not convinced about the value to society at large of paid parental leave.

We don’t have federally-mandated maternity leave in the United States, making us one of the only nations on the face of the Earth to deny our citizens this basic and vital thing.

Essentially, I disagree that it's "basic and vital."

Of course, it's nice. It's definitely a benefit for parents, and probably for their children. Likely, not so much for everyone else.

The above quote links to a Pew Research study from 2016: http://www.pewresearch.org/fact-tank/2016/09/26/u-s-lacks-mandated-paid-parental...

Nowhere in the article, or in the Pew study, do I see anything that explains to me how parental leave is a good thing for society. It's asserted as a given, without evidence, without even argument.

Consider this:

1) Having children is a choice. Especially in the contemporary world, virtually no one can say they brought a child into the world by accident. Pregnancy can be accidental. Birth is not.
2) While one is on leave, the employer must either do without a person's services, or find a temporary replacement. While no one is irreplaceable, this puts an undue burden on the employer.
3) As we move into a more automated workforce, the need for human workers is decreasing; therefore, it is not to the benefit of future society to encourage population growth, or even stagnation.
4) Likewise, the primary driver of many of the problems facing us today - pollution, climate change, hunger, maybe even war - is population growth. Reverse that, and some problems are mitigated.
5) Parental leave is unfair to those of us who have chosen not to reproduce. Yes, some organizations provide "family leave" for reasons other than childbirth, but we're talking about state or federally mandated leave, not a choice made by an employer.

On the other hand, I do recognize that in order to continue capitalism as we know it - profits driven by eternal growth - we need the population to increase. So you'd think more companies would be on board with this (more consumers = more profit). Again, though, I don't see how this could be a good thing overall.

Now, I can see how a policy of "no paid parental leave" affects women disproportionately. And that's not cool. But nothing I've ever seen has convinced me that mandated paid parental leave is a benefit to society in general (again, yes, it is a benefit to the individuals involved). The idea of its benefit is stated, as I've pointed out, without argument or evidence.

Thing is, I have an open mind. If you're so inclined, give me some good rationales and I could change my opinion. Not that my opinion matters either way, but at least I might understand the reasoning. (Anecdotes probably won't work; I've already acknowledged that it's a benefit for the individuals taking the leave.)

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