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 15 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 18 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.

2 Like

Annette in France

a day 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

2 Like

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

August 24, 2023 at 8:39am

#1054518

Cool It

A certain comedy site has trained me to see numbered lists in countdown order, and also to expect comedy from them. This, however, is not Cracked, but Mental Floss. Consequently, I'll have to supply my own comedy.

9 Creative Ways People Kept Cool Before Air Conditioning
People have tried everything from funky fan chairs to enormous ice chambers to survive a sweltering summer day.

Full disclosure: the house where I spent my childhood was located in one of the hottest and most humid parts of the mid-Atlantic (Chesapeake Bay area), and did not have air conditioning; plus, it was located about 100 meters from a massive swamp. This might have "built character," as my dad would have put it, but it did not; it ensured that I would never, ever go without air conditioning, ever again.

Not going to quote all of them here.

1. Fan Chairs

At first I thought of those movies and stage plays where the important person sits in a chair while servants wave giant feather fans. The important person stays cool; the servants get hotter. Basic thermodynamics.

People used their feet to operate the fan that moved above their head, much like someone would power an old sewing machine.

What? No servants? That de-feets the purpose.

2. Sleeping Porches

Also known as "baby disposal chutes." (For the full effect, go to the article; there are pictures.)

Another president interested in keeping cool was William Howard Taft, who had a “sleeping porch” erected on the roof of the White House in 1910.

He must have had servants reinforce the roof, or it would have collapsed. Taft made Chris Christie look like Kate Moss.

4. DogTrot Homes

The cats can just fend for themselves.

On a more serious note, one wonders why this architectural innovation is featured here, but the shotgun house is not. Despite the name, these have nothing to do with firearms, but were designed to funnel any breeze through the home's three rooms. My childhood home, noted above, started out as a shotgun.

Oh, right, because that style is associated with servants' dwellings.

5. Punkahs

These hand-operated ceiling fans have their origins in colonial India. Each year, thousands of poor seasonal workers were contracted, or otherwise compelled, to spend monotonous days pulling a cord that swept a piece of fabric back and forth across a room for the country’s elite.

AHA! I knew we'd get to the servants. "Otherwise compelled," my ass.

6. Drinking Buttermilk

EW! I'll just sit here and sweat, thanks.

The Indian subcontinent gave the world another refreshing idea for keeping cool in the searing heat: drinking buttermilk.

I guess it worked in the long run; they just landed a robot on the moon. We now have two worlds known to be inhabited entirely by robots, with just one known to be inhabited by carbon-based biological life.

Robots, incidentally, are okay to use as servants. For now.

9. Not Stressing About the Heat

How did I handle the oppressive heat and humidity when I was a kid? Well, there was a giant body of water in my front yard, and I learned how to swim.

No servants required.

4 Like • Comment • View 3 comments

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