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 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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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 1, 2024 at 9:41am

#1074643

Help Yourself

By coincidence, another Guardian link today. This one's trying to help with self-help.

Self-help was meant to make me feel better. Instead it turned toxic - and borderline dangerous

I always questioned the concept of self-help books, myself. Even putting aside the questionable credentials of some of the authors, you're not really doing self-help; you're looking to a book and an author that doesn't know you for advice. Which is fine if you're cooking and grab a recipe book.

They do, however, sometimes self-help their authors to bigger bank accounts.

I remember buying my first self-help book, which promised I could be healed of anything if only I banished my limiting beliefs.

And then if you're not healed of whatever, it's your own fault because you didn't banish hard enough!

Over the next 15 years, I bought hundreds of self-help books, courses and apps, and tracked down every self-styled personal improvement guru on TikTok and YouTube in the hope that they could teach me how to become happier, more confident and more lovable.

Apparently, the concept of "to be happier, confident, and more lovable, don't waste money on self-help books, courses, and apps; and stay away from social media" never occurred to her. Of course, none of the people involved would have given her that advice. But I can.

Look, I'm the last person who'd advise someone to "be yourself." But at some point, I realized that the way to be content with what you have is to be content with what you have.

Some approaches were gentle and perhaps even obvious: “talk kindly to yourself, get enough sleep”. Others bordered on dangerous, whether it was encouraging me to spend more money than I could afford in order to “manifest” wealth or telling me to ignore my health problems to feel healthy.

If, on the other hand, I was content with being a bigger asshole than I already am, I would totally be like, "You know the secret to happiness and eternal bliss? It's to give me all your money. Yes, all of it."

Hey, it works for preachers.

But even though I was following the guidance, nothing was helping me to feel better about myself and my brain scans still showed I was living with an incurable neurological disease. I couldn’t help but feel that not getting results was my own personal failing, rather than the industry’s with all its promises.

Yeah, that's one of the biggest problems. I mentioned it up there, too: it's easy to fall into the trap of "obviously it's my fault because I didn't pray / manifest / wish / meditate enough." Which can lead to a whole spin/rinse cycle of doom.

It took a deep conversation with a loved one to recognise that in trying to change my life, I was trying to change things that had happened to me and shaped who I was – often things that were out of my control.

And yet, she had "a loved one." I think I finally understand some of the people who are like "be grateful for what you have."

Instead of focusing on who I wanted to become, I had to quit self-help to learn and accept who I really was.

Happiness isn't having what you want; it's wanting what you have.

Since I quit my self-help crutch, I’ve realised that focusing all your energy on improving yourself can really suck the joy out of life. It makes happiness conditional: only when I have got to a certain stage can I feel contentment; only when I look that way can I be loved.

So that's your ironic self-help advice for the day. For just $19.95 a month, I can self-help you even more!

5 Like • Comment • View 3 comments

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