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 20 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.

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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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 23, 2024 at 9:09am

#1080374

Pursuit

Here's something sure to brighten everyone's day, if for no other reason than this Time article is shorter than some of the others that have popped up lately.

Obsessing Over Your Own Happiness Makes You Less Happy

Well, I'm not sure that's true for everyone. You never know what'll make someone happy, sometimes. Like, I've known people who were only happy after they made everyone around them miserable. So it's not a stretch to believe that obsessing over... whatever... might be pleasing to someone. Given all the obsessing people tend to do, I'd almost guarantee it.

Happiness is a worthy pursuit.

Is it, though? Is it really? Or are other things worthy of pursuit, and happiness is a byproduct of that?

But fixating too much on achieving it often leads to bad feelings when you fall short—which ultimately makes you less happy.

Especially if you expect not to fall short.

That’s the finding of a new study published in the journal Emotion.

For once, I actually clicked on the link provided to the study. I didn't read it too closely, but I'd take it with the same level of skepticism as any other psych study, even if it does speak to my biases.

In the study, people who said they were worried about achieving and maintaining happiness tended to have more depressive symptoms, worse well-being, and less life satisfaction than those who simply held happiness as a goal—and didn’t fret about whether they were meeting it.

Which is, honestly, exactly the result I'd expect. Which doesn't mean it's right. Or wrong. But to me it's like a study that investigated whether cumulus clouds look soft and fluffy.

What’s the secret, then? Take the pressure off and stop taking your own happiness temperature so often, Zerwas says.

Okay.

Embrace all of your feelings—both happy and sad ones—since all emotions can be informative, providing us insights into our psychic makeup.

Okay.

And practice cognitive-behavioral strategies such as mindfulness—being present in one’s emotions and aware of what those feelings are—to truly tune in.

Aaaaand you lost me.

I feel like this "pursuit of happiness" thing is mostly an American phenomenon, thanks to the famous words in the Declaration of Independence. There's some discussion

over what exactly was meant by "happiness" therein, but I'm pretty sure the connotation of the word has changed in the last 250 years. It probably didn't mean a feeling of unbridled joy, but something closer to security.

And in my experience, living without fear is happiness enough.

4 Like • Comment • View 2 comments

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