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 9 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 12 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 23 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!))

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November 4, 2023 at 11:44am

#1058881

Brew-tality

Inspired by "Invalid Item"

, though this particular observance wasn't in the suggested prompts. The first Saturday in November is officially:

Learn to Homebrew Day

One question I get asked on a regular basis, once someone discovers that I'm a beer snob (which happens pretty quickly, but not as quickly as some people tell you "I'm a vegan."): "So, do you brew your own beer?"

No.

Not just no, but HELL no.

I mean, sure, the process is fascinating, and as with most work, I could watch others do it all day. And I'm glad some people are into that, because that's how we get brewers at craft breweries. So, of course, being curious, I looked into it, and discovered that home brewing is approximately 1% enjoying the product; 3% brewing; and 98% cleaning and sterilizing.

Yes, I'm aware those add up to >100%; I'm using hyperbole to emphasize just how much I hate, loathe, and despise cleaning.

Why do all that work when I can go to a bar or brewpub and, using the fruits of my other labors, purchase a delicious golden beverage that's the result of other peoples' labor? It's not like diamonds, coffee, smartphones, clothing, avocados, or chocolate; you can be fairly confident that craft beer in the US isn't brewed using slave or child labor.

Still, I wanted to take a moment to appreciate that here in the purported Land of the Free, we have the freedom to brew our own beer, if we want to put in the really outrageously tremendous amount of work involved. It wasn't always that way.

I'm not just talking about those dark times known as Prohibition, either. Once Prohibition was lifted—an anniversary I also celebrate, coming up next month—it took a while before home brewing was legalized.

By "a while," I mean "nearly half a century."

Before Prohibition, brewing had mostly been done at home, though industrialization facilitated commercial breweries more and more as time went on. Many, if not most, of the Founding Fathers of the US were brewers (or their spouses were; brewing, like cooking, was considered largely a woman's job). The entire idea of the American Revolution was fomented (as opposed to fermented) in pubs, so beer is in our DNA, which made Prohibition that much more of a slap in the face to Lady Liberty.

Amendment XXI to the US Constitution, passed on December 5, 1933, repealed the earlier amendment XVIII that created Prohibition. Yay. But home brewing didn't become legal until 1978, 45 years after the repeal of Prohibition. Thanks, Jimmy!

Which is not to say that people didn't break The Law on a regular basis, but the legalization of homebrew led inexorably to our craft beer landscape, which remains the primary reason I travel. Last I checked, which was five minutes ago, there were nearly 10,000 breweries in the US. I'm not even sure how many of those I've visited and/or sampled brew from; my records are spotty, as you might expect after I've had a couple. It's also complicated by the fact that, on average, every week, we lose one and gain two.

With that much beer to choose from, why bother making my own?

No, if I'm going to do work, it's going to be important work, like figuring out how to visit the 9000 or so that I still haven't.

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