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

1 Like

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

August 26, 2024 at 10:30am

#1075731

Literally the Opposite

Today, we have a bit of linguistic education from that well-respected teaching site, Cracked:

5 Words That Switched Meaning Because Everyone Used Them Wrong
‘Entitlement’ now means you don’t deserve something, which is absurd

There are way more than five, but we have to take tiny attention spans into account, here.

Some words mean two things that are complete opposites. “Cleave,” for example, can mean both splitting something apart or joining two things together.

This is known as a contronym, and I've discussed them before: "Contronyms"

. But that was almost five years ago, and today's bit isn't exactly about contronyms.

5. ‘Steep Learning Curve’

And already, we step out of theme: that's a phrase, not a word.

A steep learning curve indicates a process that’s easy to learn, with production rates rapidly increasing over time. With a shallow curve, on the other hand, rates increase more slowly, indicating a process that’s harder to learn.

Though, as the article so helpfully points out, a lot depends on how you define your graph's axes.

Ask me, it's probably best to do as they suggest and stay away from "learning curve," especially since that borders on corpspeak.

4. ‘Mystery Box’

If you’re a normal person, you may go your whole life without hearing about mystery box storytelling. If you’re a nerd, however, you know the term well. Mystery box storytelling is when a series lures you in by dangling a bunch of mysteries, mysteries to which the writers themselves do not yet know the answers.

Well, I'm a nerd, and I think I know a few things about writing, but I'd never heard that particular phrase. (Again... phrase, not word.)

The name “mystery box” comes from a TED talk that J.J. Abrams gave in 2008, initially inspired by people asking him what the island in Lost is.

The real mystery box here is how one director can fuck up so many different franchise movies, while still continuing to get hired as a director.

If I were inclined toward conspiracy theories, I'd say Paramount (the owner of Star Trek) hired him as a double agent to ruin Star Wars for Disney.

It’s a bit odd, though, to call them mystery box shows after Abrams’ choice of theme for one TED talk. Because he used the box to talk about so many different concepts, and because mystery boxes are an actual thing, with one specific mystery: What’s inside, and is it worth the price you paid?

But the point here isn't to slam Abrams as a director (I liked a lot of his stuff, just not all of it); it's to show how the phrase got to be misused, and for that, there's lots more background in the article.

3. ‘Entitlement’

Hey, look! A single word, unattached to others!

An entitlement is something you’re entitled to. That’s what it means. That’s how nouns work.

In the U.S., federal benefits programs are called “entitlements,” based on the assumption that recipients are entitled to them. Some of these programs, like Social Security, are contributory, which means you personally paid into them. Others, like nutrition assistance, are non-contributory, paid from the government’s discretionary budget, but by calling them “entitlements,” the government is still saying recipients are entitled to them.

This has bugged me for years: people calling contractual or legal obligations "entitlements" like it's a bad thing.

We’ve now reached the point when advocates for entitlements argue it’s insulting to refer to these programs as entitlements:

Shit, I've been there for at least a decade. Maybe longer. Maybe less. My relationship with duration (durationship?) can be tenuous.

2. ‘Low Man on the Totem Pole’

And we're back to phrases.

You probably think the “low man on the totem pole” is the most junior person in an organization. They have not ascended very far, and they lack power. But have you considered what sort of person would manage to be the bottom figure in a human totem pole? They would need to be strong enough to support everyone else.

Well, workers *are* the most important people in an organization.

In a totem pole, the bottom figure isn’t some peon being squashed by the important figure on top. They’re the figure in the pole who’s been carved with the most detail, and they may also be the biggest.

Eh, I avoid the phrase anyway, for cultural reasons. But now I'll never see it in the same way again.

1. A Special Note on ‘Literally’

Oh, hell, I'm going to hate this part.

The dictionary now notes that the word has a second definition, which essentially means “not literally,” and this inclusion attracted a lot of controversy online around a decade ago.

"The dictionary" (misleading, as there are several entities competing in that space) is descriptive, not prescriptive, at least the English ones. It pretty much has to base its definitions on how people are using words... even when that usage is objectively wrong, as with emphatic uses of "literally" or any use of "decimate" to mean something other than "remove one-tenth of."

Literally doesn’t serve here as a synonym for figuratively, a disclaimer clarifying that they’re joking. It serves as emphasis. In fact, it emphasizes their statement by declaring it’s literal. You know it’s not literal (again, from context), but they’re telling you it is, and therein lies the sentence’s humor.

Okay, so, by framing it as comedy, you're moving me to your side. Waltz's Second Law states: "Never let the facts get in the way of a good joke. Or a bad one. Especially a bad one."

So, people generally don’t use “literally” as a synonym for “figuratively.” But they used to.

Go through classic novels from a century or two ago, and you’ll find many authors doing so.

Huh... okay.

Despite the criticisms that young people misuse “literally,” using it as a replacement for “figuratively,” doing so now comes across as a very formal or outdated way of speaking. The word did change meaning, but it reversed in the opposite way to what people think, becoming more literal.

And there it is, folks: proof that I can be persuaded, given enough evidence. And comedy.

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