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.

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

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.

2 Like

October 15, 2021 at 12:06am

#1019364

How the Bible Can Lead You Astray

Yeah, I know the entry title is clickbait. Hey, it's a legitimate writing skill.

4 Words You're Using Wrong (Because Of The Bible)

Of course, the actual article is from Cracked, and it's only about individual words which, of course, only have whatever meaning we decide they have at any particular time. Kind of like everything else, actually.

The Bible is how we got such words as "spake," "mammon," and of course "Nehemiah."

Spake -- wasn't he the cousin Spock never talked about?

Also surprised they don't mention Nimrod in this article. You've probably called, or heard someone call, someone a "nimrod" to mean something like "idiot" or "fool." Nimrod was described as "a mighty hunter" in the Bible; Bugs Bunny called Elmer Fudd "Nimrod" ironically and boom, a new usage is born. Praise Bugs!

It also contributed some words that we actually use.

Well, actually...

Apart from a boatload of names, the Bible didn't directly contribute a lot of words; the English translation of it did. Of course it's translated into hundreds of languages now, but the original was written primarily in ancient Hebrew, Greek, and Aramaic. But I think the article is mostly taking about the KJV, which uses language similar to Shakespeare because it was published at about the same time.

4. There's Nothing "Immaculate" About Virginity

It seems a lot of people don't know what "immaculate conception" means. Because the Immaculate Conception was not the birth of Jesus to a virgin (an event you can instead conveniently just refer to as "the virgin birth"). It was the conception of Mary herself. Mary's mother was no virgin -- Mary's parents had sex, which is the normal way of making kids -- but the conception of Mary is called "immaculate" because she herself had no original sin.

To further complicate matters, "immaculate" is a Latin-root word, and you'll note that Latin isn't in the list of languages in which the Bible was originally written. It was, of course, translated into Latin very early on. Bonus knowledge: "macula" in Latin meant "spot," in case, you know, you were looking for a name for your dog.

Unborn babies aren't generally known for sinning much, especially if you revere unborn children as much as the Catholic Church does. And yet, an immaculately conceived baby is actually one whose parents had sex just fine but whose own soul is free of sin. Free of original sin, the flaw every other baby starts out with, ever since those ancestors sinned back in Eden.

Look, I like Cracked, obviously, but they're not theologians; they're comedians. If the sentences I just quoted seem to contradict themselves, they do; welcome to the world of theology.

But rest assured, immaculate conceptions have nothing to do with abstinence, and if you yourself were conceived thanks to good old copulation, there's nothing dirty about that. Religions may have some strange ideas about sex, but no religion says it's dirty for a husband and wife to have sex to make a child (least of all Catholicism, which is super into marital sex). Any religion that does claim that will have some trouble creating new followers.

There have absolutely been religions that say this, and in accordance with the prophecy, they had trouble creating new followers.

In any case, as we've seen numerous times (see also "decimate,") the meanings of words and phrases change over time, so unless you're a Bible literalist, at some point you're going to have to go with the flow and accept that Anakin Skywalker had an immaculate conception.

3. There's Nothing Especially Good About Samaritans (Though Also Nothing Bad)

This one has bugged me for a while, because unlike mere words, "Samaritan" denotes a particular group of people, particularly the group living circa 30 C.E.

The most famous parable in the Bible is surely the one about the Good Samaritan, the guy who helped a battered robbery victim after various other travelers turned a blind eye. And so today, anytime someone does a good deed, you might call them a "good Samaritan." Or, just call them a Samaritan -- "good Samaritan" feels redundant. There's even a charity simply called "the Samaritans."

This is like calling your group "the Edomites" or "the Third Dynasty Egyptians."

Samaritans were an ethnic group. Still are an ethnic group, actually, which is why it's kinda weird that we use the word to refer to anything other than the real people today.

Okay, so their descendants are in fact still around. Yes, I learned something.

In a Biblical context, Samaritans were mainly relevant for being an ethnic group Jews didn't much like.

I think a modern parallel would be how a lot of people in Europe feel about the Roma or... well, you know, the Jews.

The reasons for this prejudice were long and complicated, so all you need to know is that, as with every interethnic political feud, there was never any good reason to hate individual members of the maligned group.

I'm just leaving this quote here because it's vitally important to remember.

It would be like if, today, someone teaches a lesson by telling a story about three men locked out of their homes. The first man, a priest, climbs up a drainpipe to an unlocked window, slips, and breaks his neck. The second man, a rabbi, drives his pickup truck straight through the closed garage door, causing much property damage in his quest to gain entry. But the third man is Florida Man, and despite what you'd think, he phones a locksmith and has his door opened professionally. This is the story of the Prudent Florida Man, a lesson that subverts our expectations.

That's... almost... a good joke, which is in fact what one would expect from a comedy writer.

As years go by, now imagine people start calling various prudent people "prudent Florida Man," and then simply "Florida Man," because that's what they now think “Florida Man” means. That means the parable does a hell of a good job at breaking the original Florida Man stereotype, breaking it so hard it's eventually forgotten. But it also forgets half the original message of Jesus' parable, which is "just in general, don't be racist."

And again, just, you know, leaving this here.

2. Every Prodigal Son You Can Think Of Probably Isn't Prodigal

Over the years, people often quipped "the prodigal son returns" when someone returned, so "prodigal son" itself became a phrase that means "one who returns." It's funny how often that sort of things happens with language. Like how we said "good Samaritan" so much, we started thinking "Samaritan" means "someone good." Or, say, what happened with the phrase "damsel in distress." We used the phrase "damsel in distress" so much, we started thinking "damsel" itself means "woman in distress." Some people even now use the word damseling, when they want to say turning a woman into a woman in distress. And yet damsel really just means "young woman."

As an aside, so does the word in Isaiah 7:14 which is usually translated as "virgin." That particular verse wasn't a prophecy; the writer of Isaiah was basically saying "when pigs fly."

But anyway, back to prodigal. What does prodigal really mean, you ask? It means "spends wastefully."

Meanings change, yes, but I do think it's important to know how they change.

In fact, in the news media, "prodigal son" is often misused for someone who leaves home and makes a lot of money, which might be the opposite of being prodigal.

Not the first time a word has come to mean its opposite, and likely not the last. See also: cleave.

1. "Vengeance Is Mine" Means Don't Seek Vengeance

Here's another phrase that appears lot in pop culture. "Vengeance is Mine" is the title of a dozen different movies, as well as TV episodes, books, and songs. Always, the title comes from someone getting revenge. Take the classic Roald Dahl story "Vengeance Is Mine, Inc." Two broke bothers (prodigal ones perhaps) decide on a moneymaking scheme: taking revenge on clients' enemies, such as by stripping victims to their underpants and then dumping them on a public street. We see them carry out one job, breaking a newspaper columnist's nose, and then they ride off into the sunset.

I mean, okay, I admit to being misled by most of these, but I always knew about the "...sayeth the Lord" completion of that particular Bible quote.

Which makes sense. Most religions believe sinners get punished after death. Well, maybe not Judaism, so there could be room for vengeance if you take an Old Testament view, but other religions preach some form of hell, or karma, or reincarnation. That means religious people should see no need for exacting vengeance on those who are still alive, since that'll be taken care of supernaturally in the afterlife. And yet, it's often religious people who most believe that it's our duty to punish the wicked.

No, that is not what most Jews believe about vengeance, and thinking that they do contributes to centuries of Judaeophobia (a word I coined because everything has to have a -phobia ending these days, so let's be inclusive here), so stop that.

There are other reasons for punishment (deterrence, rehabilitation, incapacitation), but it's retribution (vengeance) that so many moral crusaders demand when they call for harsh sentences. Well, the next time you hear a Bible thumper say we need to fry a criminal because it's what they deserve, ask, "Have you so little faith in our Lord in heaven, that you doubt He will punish evildoers as He sees fit?"

On second thought, don't say that. That's how Jesus talked, and we all know what they did to that guy.

So there you have it, and yes, sometimes it pays to get your theological instruction from a comedy writer.

On a personal note, I think I've fully recovered from my procedure yesterday morning, and everything went well. So, bad news, you'll probably have to put up with me for a while longer.

Probably.

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