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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 6 hours 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.

1 Like

🐕GeminiGem🎁

about 10 hours ago
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

about 12 hours ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

about 13 hours ago
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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2 Like

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

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 8, 2024 at 9:10am

#1079661

Bungalow Bash

Today, we have an exercise in metaphor-stretching:

Caveat Bungalow
The ostracizing of normal

This starts out looking like a piece about housing:

When I was a kid, I lived in a bungalow for a while. You know, one of those houses that’s compact and square and has a half-upper floor that’s basically just a loft with sloped ceilings.

Yeah, well, the house I lived in as a kid was originally a shotgun shack.

Now if someone told me to live in a loft with sloped ceilings I’d ask if there was any regular house available, please.

Nice to be able to be picky, isn't it?

Something else I’ve noticed that’s shaped like a bungalow is the way we talk to each other. Human speech.

And that's the metaphor part, which, as it turns out, is what the post is really about.

We waste an unbelievable amount of time — in our daily lives, on podcasts, in interviews, in blogs and articles, even in Tweets and Notes for God’s sakes — qualifying everything we say with caveats. We say “now I know not all x are y, and I know that historically abc, and I’m not trying to say that lmnop so please don’t take this wrong…”

Granted, I avoid X/Twatter like the plague it is, and I don't listen to podcasts. But while I've seen what this dude's talking about, and even engaged in it on occasion (for instance, when I note that I might be talking about something with a US bias), I don't think disclaimers like that rise to the level of wasting "an unbelievable amount of time."

Saying anything even remotely controversial on the internet is terrifying.

Is it though? Is it more terrifying than saying it in a crowd of Americans with guns?

No matter what argument you make on the internet, you will get people who reject it wholesale because you forgot an asterisk. Because you forgot to mention their particular edge case.

Yes, and?

We all know that not all heterosexual dating advice or sex advice applies to transgender people. It doesn’t even apply to all straight people. We are all intelligent enough to know that.

[citation needed] on the "intelligent" bit.

People have managed to make it insensitive to speak about things that the majority of people deal with. People have managed to make it insensitive to be normal.

And what, exactly, is "normal?"

And then the tyrannical minority ostracizes you for it, and in turn makes it okay for everyone else to ostracize you.

"Tyrannical minority?" Loud, sometimes, maybe, but that seems like an oxymoron. Unless the "minority" is an actual, political tyrant, a minority of one.

If you drove down a street consisting entirely of overly-decorated bungalows, with nice upper windows and big, furnished porches, you’d call bullshit on the entire street.

No, I don't think I would.

Everywhere I look, I see people decorating their speech with nuance when all they really want to say is some simple, normal thing. It feels like bullshit.

Well, it's not everywhere I look. Perhaps examine your own biases, first?

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.

Some people are just actively searchlighting for reasons to get outraged. They aren’t worth listening to.

On that, we can agree.

People who need that much nuance weren’t going to learn from your argument anyway.

Aaaannnd you've lost me again.

As I've said before — we as a culture have become profoundly unserious.

To me, the opposite of "serious" is "funny." So the opposite of "unserious" would be "unfunny." Me? I'd rather be funny.

Maybe if we just start saying what we mean and placing the impetus on the reader to read nuance into the topic, we’ll all grow up a little.

Well, sure, calling me childish certainly helps your argument.

People are tired of having to decorate their speech to make it marketable.

I think by "people," he means "I." As in him, not me.

I think we’re past peak Woke, and I think part of what that means is that we’re past peak not-being-able-to-speak-like-adults.

Ah, there it is.

I'm not dismissing his argument, mind you. I read the whole thing, top to bottom (it's really not that long). I simply don't agree with most of it, though I accept that my opinion could change.

If you're going to do a metaphor, make sure it's one that we can actually relate to. Bungalows may not be exotic or ritzy, but they're generally better than no home at all.

So that's me, saying what I mean.

4 Like • Comment • View 2 comments

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