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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 21 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 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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