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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QPdoll

about 7 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 10 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

about 21 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

2 Like

December 26, 2023 at 9:07am

#1061420

Biggerer

Another Cracked article, once again illustrating the randomness of my random number generator; I have far more links from other sources.

5 Things Getting Bigger When No One’s Looking

I am resisting the obvious joke. Resisting with all my might. Resisting hard, you might say. Can this article also resist?

You turn around and see something behind you. Later, you turn around again, and it’s bigger now. This means either it’s following you, or it has physically grown in size.

Or both. Why not both?

5. Spiders

You'll want to click on the link for the illustration of what a spider might look like.

Bigger spiders help cities, since bigger spiders can gobble more bugs. They also serve as more food for birds. Of course, any change in the food web can spark unpredictable consequences, and spiders eating too many bugs or feeding too many birds could be disastrous, but a thorough analysis of the side effects leads the scientists to conclude that, yes, giant spiders are good news, everyone.

Spider-like typing detected.

4. Eyes

Including spider eyes?

One part of your body grows for your entire life. We’re not talking about fat — many parts of your body grow when you eat more, but one part grows regardless.

Almost did the obvious. Almost.

We have one single confirmed example of an organ that grows, forever. It’s the lens of your eye. Each lens grows by 1.38 milligrams a year, and while that sounds like nothing, we have to take into account just how small lenses are to begin with. Between early adulthood and when you die, you can expect each lens to double in size.

Unless, of course, you get off the growth treadmill by having cataract surgery. Which reminds me, it's been two years now.

3. Finland

As sea levels rise, countries all over the world have to reckon with losing some land. This loss will range from an inconvenience to apocalyptic.

Nah, they'll just redefine the border as being 100 yards out from shore.

Finland is an exception. Finland gains 7 square kilometers (3 square miles) a year. No, it’s not because they’re so militant about combatting climate change that they’ve somehow convinced the sea to recede. They’re instead experiencing a phenomenon called post-glacial rebound.

The opposite of shrinkage?

2. Breasts

...

...

...

I'm sorry, did you say something?

1. Zoo Animals

Maybe the animals are under the influence of endocrine-disrupting chemicals, or maybe they’ve been infected with some obesity-causing virus. Maybe that’s also why we’re getting fatter, rather than the traditional explanation of overeating.

But that explanation offends puritans, who are certain that everything from obesity to climate change is our fault because we've sinned.

And there it is... the whole article and my commentary, without the obvious joke. I feel shafted.

6 Like • Comment • View 6 comments

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