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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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.

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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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

March 19, 2019 at 1:16am

#954570

Good and Gooder

https://qz.com/1562585/the-seven-moral-rules-that-supposedly-unite-humanity/

An Oxford researcher says there are seven moral rules that unite humanity

Well. Okay. He's from Oxford, so he has to be right. Right?

In 2012, Oliver Scott Curry was an anthropology lecturer at the University of Oxford. One day, he organized a debate among his students about whether morality was innate or acquired.

That sounds more like a philosophy question than an anthropology question - though I suppose there's some overlap. Either way, I wonder if it matters. I guess that's a philosophy question, too.

Religious people, in general, insist that morality comes from God (or whatever higher being depending on your religion). Atheists are pretty sure it doesn't (after all, there is no god but morality is a thing that clearly exists, in the way a lot of abstract concepts clearly exist). I've even heard the argument, "But if you don't believe in God, then what's to keep you from stealing, raping, and murdering?" To which I reply something along the lines of, "Hey, if the only thing keeping you from stealing, raping and murdering is that you believe God told you not to do those things, then please, please, keep believing."

Of course, belief in God has never stopped certain people from stealing, raping, and murdering. Neither has disbelief. This still doesn't answer the question of innate morality vs. acquired (nature vs. nurture), or whether it matters.

So back to the article.

Morality, he says, is meant to promote cooperation.

Ooooh, nice weaselly passive voice there. Who meant it to promote cooperation? God? Or the humans who came up with the rules? Or is it a survival thing that goes hand in hand with evolution?

To be fair, later there's a statement talking about how "morality evolved to promote cooperation."

Now, I'm not going to list the Seven Cardinal Virtues Universal Rules of Morality. That's what I put the link in there for. (I double-checked this time. It's there.) But I will note my extreme skepticism about how the rules' parameters were codified. For example, "Help your family" and "help your group" seem awfully similar to me.

But then, I'm not an Oxford professor.

I'll just close with a couple of examples of how expressions of morality change over time.

First example: slavery. I think you'd have to work hard to find someone who believes that slavery is morally right, these days, and even harder to find someone who would admit to that belief. And yet, two hundred years ago, you could find people on both sides of that issue. And here in the US, there were people on both sides who pointed to the very same (supposed) arbiter of morality - the Bible - to support their beliefs. So what changed? Well, Cynical Me says the only thing that changed is industrialization reducing the demand for cheap labor - and Cynical Me is pretty much Me - but at the same time, it became an issue of morality.

Second example: child-rapers. Again, find me someone who thinks this is a morally good practice. Oh, sure, you might find arguments about what ages constitute being a child, but I think most of us can agree that it's a terrible idea. And yet, there is, in the Bible, not one commandment along the lines of "Thou shalt not fuck children." This is morality that we definitely came up with all by ourselves - so I'm pretty sure we came up with the other rules all by ourselves, also.

Anyway, I could go on, but what would be the point? Mostly I just wanted to share the article because it's interesting and worth discussion.

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