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 20 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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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

August 18, 2024 at 10:39am

#1075432

Revisited: "Challenge Complete"

Today's cruise on the river of time takes us back to the end of January, 2019. This was a 30 Day Blogging Challenge month, and for context, my current daily blogging streak wouldn't start for another ten and a half months. And, of course, it was before everyone's lives got disrupted.

The entry itself, "Challenge Complete"

, featured the kind of prompt I came to expect for the final entry into that challenge:

Write a list of at least 5 blogging prompts to add to the Challenge War Chest to be used for future rounds of the 30DBC. Then, use one of your own prompts to write your entry.

I've never really enjoyed coming up with prompts. Scratch that; I do enjoy it; it's just that when I'm expected to create one, my mind goes blank. I know this sounds odd from someone who's been judging "Invalid Item"

and therefore coming up with a prompt almost every week for nearly two decades, but there you have it. I just think of it as another challenge for myself.

The actual prompt list I came up with is uninspiring to me even now, and I won't quote it here. It's there at the link if you want. I'll just address some of the other things I said in the entry.

I don't know who coined the term "bucket list," and I can't be arsed to find out.

Unlike most words and phrases, they've traced that phrase in its "things to do before you die" sense to a precise moment.

I do know that it became the title of a moderately interesting movie with Jack Nicholson and God.

According to that link I just did, the phrase's origin and the movie title came from the same place.

While I've been known to use the phrase from time to time, I usually call it my "fuck-it list." This is because I've already done most of what I set out to do, and have given up on the rest.

Good to see my attitude hasn't changed in five and a half years.

It's come to my attention recently that life is much less frustrating if you can separate "things that I cannot control" from "things that I can control," and to focus on the latter while letting go of the former. This is something akin to the infamous Serenity Prayer, which I resent because a) it's a prayer and b) it's associated with AA, and AA is for quitters.

That's not my actual problem with AA, but it made a decent joke.

I've been walking every day for the past 5 weeks, which is, like, a new record for me for exercise.

I should have kept track of my record for "days without formal exercise." It'd be a lot longer.

Say I want to take a trip to space. How can I make that happen? Well, probably by paying Virgin Galactic a very large sum of money to fire me into a suborbital trajectory that technically takes me 100km up for a fraction of a minute, thus putting me, by definition, in space.

I think that particular venture never took off (pun intended). Branson lofted himself into a suborbital trajectory, but his maximum altitude was somewhat less than the accepted boundary of "space." Pretty sure I wrote about it in here at the time. Then Virgin declared bankruptcy. Or something. I can't keep up with all the rocketing these days.

Honestly, even if I were a billionaire, this would strike me as a colossal waste of money.

While I accept that other people wouldn't find it so, paying any amount of money for fame and bragging rights.

So here's the real example: I want to be published.

Yeah, not so much anymore.

Not much else of note. Kind of a boring entry, even now. Only one joke that I could find, and it wasn't even original.

Can't be "on" all the time, I suppose, but some challenges are never complete until the day you kick the bucket.

1 Like • Comment

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