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.

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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.

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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.

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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.

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October 14, 2020 at 12:06am

#995840

Morality Play

While I try to stay away from the Big Three subjects guaranteed to ruin a good relationship (politics, religion, and Monopoly), sometimes my amateur interest in philosophy intersects with one of them.

This is one of those times. I saved this link a few months ago; it's only now come up at random, and I'm completely sober (for the moment), so fasten your seatbelts and let's go.

https://aeon.co/ideas/how-a-huguenot-philosopher-realised-that-atheists-could-be...

How a Huguenot philosopher realised that atheists could be virtuous

I admit to having, for most of my life, only a very vague idea of what a Huguenot was. All I knew for sure was it was a religion thing, and not in any way related to an astronaut. Fortunately, the internet came along and I obtained a better idea. For those who are wondering and can't be arsed to look it up, they were French Protestants in the 15th-16th century who were persecuted for their religious beliefs.

Sound familiar? It should.

Here's a bit about their history.

Anyway, the Aeon article.

For centuries in the West, the idea of a morally good atheist struck people as contradictory.

I'd even go so far as to extend that from "atheist" to "anyone who didn't follow your particular brand of religion."

But today, it is widely – if not completely – understood that an atheist can indeed be morally good.

Widely? [Citation needed].

One of the most important figures in this history is the Huguenot philosopher and historian, Pierre Bayle (1647-1706).

I'm just leaving this here because, unlike certain websites, I don't tease and then fail to deliver. That was the philosopher mentioned in the headline above.

Bayle introduces his readers to virtuous atheists of past ages: Diagoras, Theodorus, Euhemerus, Nicanor, Hippo and Epicurus. He notes that the morals of these men were so highly regarded that Christians later were forced to deny that they were atheists in order to sustain the superstition that atheists were always immoral.

Sounds like a variation on the No True Scotsman fallacy

. "No true Scotsman puts brown sugar on his porridge. The fact that Angus MacGregor puts brown sugar on his porridge just proves that he's no true Scotsman!"

From his own age, Bayle introduces the Italian philosopher Lucilio Vanini (1585-1619), who had his tongue cut out before being strangled and burned at the stake for denying the existence of God. Of course, those who killed Vanini in such a fine way were not atheists.

"It is wrong to kill. Anyone who disagrees with us on any other point probably disagrees with us on that one, too, so we must kill them."

The really pressing question, Bayle suggests, is whether religious believers – and not atheists – can ever be moral.

Perhaps it was a pressing question for Bayle in the 17th century, but it's clear to this atheist that, certainly, religious believers, as well as atheists, can be moral. You mostly just have to watch out for individuals of whatever stripe who want to do wrong things, and for believers (or unbelievers) in larger groups.

Left alone to act on the basis of their passions and habitual customs, who will act better: an atheist or a Christian? Bayle’s opinion is clear from the juxtaposition of chapters devoted to the crimes of Christians and chapters devoted to the virtues of atheists.

And, clearly, counterexamples of both sorts (and from other religious groups) can easily come to mind.

In Bayle’s time, to be truly good was to have a conscience and to follow it.

Here's an interesting philosophical puzzle I've been turning over in my head for some time. I'm sure if I had more than an amateur interest in philosophy, I could find some proposed answer to it out of some grizzled head or other, but I haven't run across one yet.

Suppose you have two people, let's call them Alice and Beth. Alice has a strong desire to do a particular immoral thing. Let's keep this simple and say, "shoplifting." So Alice really, really wants to kife things from stores and take them home without paying for them. But she doesn't; she restrains herself because she knows it's wrong, or maybe simply because she doesn't want the hassle involved with getting caught. Doesn't matter; the point is that she controls her urge. Beth, on the other hand, has no such desire; she understands that shoplifters exist, but she has no desire to do it, herself. Very, very different motivations here -- but the result is the same: neither Alice nor Beth engage in stealing shit from stores.

The puzzle is this: Between Alice and Beth, who is more praiseworthy? Alice, for keeping her inner monster at bay? But she acknowledges that she has that inner monster, that desire to do something wrong, and is that not itself evil by some definition? Or is it Beth? But the reason Beth doesn't shoplift is because she has no desire to do so; her motivations are "pure." Consequently, why should that be praiseworthy? She's just following the dictates of her nature. And yet... the outcome is the same for both of them: no shoplifting. An external observer only sees the result, not Alice's inner turmoil or Beth's lack thereof. Unlike with Cindy, who doesn't have a job and really needs food so she occasionally swipes a few apples from the grocery store. Or Debby, who just does it for the thrill of it.

For the sake of this line of thought, assume all four of these ladies have similar religious beliefs (or lack thereof, whatever).

The point, as regards this article, is that merely following one's conscience doesn't always lead to an outcome that we external observers would consider moral. Or immoral.

The challenge Bayle undertakes is to explain how atheists, who do not recognise a moral cause of the Universe, can nevertheless recognise any kind of objective morality.

Well, that's easy enough, I think. What are the effects of your actions on the rest of the world, and the people around you?

This article limits itself to Christianity and atheism, but of course myriad other religions exist. Christianity is itself a form of atheism: it rejects all of the other gods ever imagined by humankind. As do I. The only real difference is that I believe in one fewer god than they do.

God cannot make killing innocent people a morally good action. Respecting innocent life is a good thing that reflects part of God’s very nature.

"Therefore, we declare that anyone who disagrees with us is not innocent, et voilà, we can kill them." Also, see the book of Joshua. "Those people have the audacity to be living peacefully in the land I think God wants us to have. To the siege engines!"

At bottom, these Christian views do not differ from what atheists believe about the foundation of morality. They believe that the natures of justice, kindness, generosity, courage, prudence and so on are grounded in the nature of the Universe. They are brute objective facts that everyone recognises by means of conscience.

Okay, look. That's nice to hear and all, and I'd like to think it applies to me, but any sentence that begins "Atheists believe..." should be rejected out of hand. The nature of atheism is to accept that there are no gods. It says nothing about what we believe, only about what we do not believe. Personally, I'm convinced that these enumerated virtues, and others, are a result of human evolution, a kind of social lubrication because we are social animals. They are traits that help us get along with each other, to be able to live among others. And, naturally, they're ideals that few of us humans, religious or otherwise, always live up to; that's the way "ideals" work. Of course, dig deeper and yes, human evolution itself is grounded in the nature of the Universe, so I agree with this assertion for myself - but I don't claim to speak for anyone else.

According to the Canadian philosopher Charles Taylor...

"Canadian philosopher?" That's a phrase I've never heard before.

...our age became secular when belief in God became one option among many, and when it became clear that the theistic option was not the easiest one to espouse when theorising about morality and politics. Through his reflections on atheism over three decades, Bayle demonstrated that resting morality on theology was neither necessary nor advantageous.

And just to be clear, because I'm not sure I have been, I'm not trying to convince anyone over to my point of view, or to disrespect anyone's religious views. As I noted above, this article focuses on Christianity because it's about a philosopher who lived in a Christian country during a time of religious upheaval and persecution, but similar arguments can be made with any belief system. My goal here is only to expand on the article itself, and present my own point of view. But part of my point of view is radical freedom of religion: you practice yours (or not) and I practice mine (or not), and we can still try to get along. Yes, religious people and atheists have both committed atrocities in the past. That doesn't mean we have to continue to do so.

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