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 8 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🎁

a day 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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day 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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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

September 11, 2024 at 11:48am

#1076615

Drop and Gimme Twenty

Somewhat predictably, I'll take a break from the usual today in order to blab about this being my 20-year anniversary here.

Unfortunately, I said most of what I wanted to say back on September 1, as we began the site's 24th anniversary: "Four and Twenty"

Having my own milestone so soon after that of the website itself, though, is kind of like I'd imagine it would feel like for your birthday to be on December 26: everyone's burned out on celebrating, including me. Especially me.

Which is fine. I'm on board with celebrating actual birthdays, but membership anniversaries are just, well, whatever. Still, this being the 20th, one of those nice rounded-end numbers, I felt I should say something about it.

Things were a bit different back in 2004. Smartphones weren't ubiquitous yet. The internet hadn't become a mire of scams, ads, and influenzas. And a certain other event that took place on a September 11 was still fresh on everyone's mind.

I hadn't intended to join on the 11th, for that reason. I tried to create an account here the previous day (my dad's birthday, and he was still alive back then, so already a significant day for me), but for whatever technical reason, it never took. That's also how my username got the annoying 02 at the end; while the account didn't work, my attempt was enough to block off just "cathartes" as already in use. It's been long enough that I don't remember all the details exactly (hell, 20 days is probably long enough for that, for me), but I vaguely recall creating this account with help from a friend who was already a member, shortly after midnight, forcing my account anniversary to be on a day that was already infamous.

I've considered changing my username to something else, but never went through with it. It's kind of part of my identity now, so I'm ambivalent about the idea.

So, I'll just finish by posting what I'm pretty sure was my first item on here, created on September 11, 2004—but which I'd written some years earlier: "Ghost Poem #1"

[ASR]

Odd that it was a poem, I suppose, because I'm more of a fiction writer and, now, a blogger, but I suppose it was something quick and simple to put up here to get started. And even back then, when I had an actual job and a spouse, I was all about quick and simple.

Some things don't change all that much.

6 Like • Comment • View 4 comments

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