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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Wordsmitty ✍️

about 2 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.

1 Like

Annette in France

about 13 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

about 17 hours 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

1 Like

WakeUpAndLive️~Happiness

about 17 hours ago
In response to "Paris, I Guess"

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

1 Like

Elisa: Middle Aged Stik

about 22 hours 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

about 23 hours 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

about 23 hours 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

a day ago
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

Cubboo!

a day ago
In response to "Get Baked"

Yes! I discovered this a few years back (slitting the potato about 1/4 - 1/2" lengthwise, then one or two slits across the longer slit.) It came out wonderful! It seems like this was called an English jacket potato, or something like that.

2 Like

April 15, 2024 at 8:25am

#1068809

That's Cole

Not to brag, but I've known this for a long time. Still, there's always new stuff to learn. This is a fairly old article from Vox:

Kale, Brussels sprouts, cauliflower, and cabbage are all varieties of a single magical plant species

And collard greens. Also kohlrabi, but every time I mention kohlrabi people be like wut?

In some circles, kale has become really, really popular.

Too popular. I think people may have finally calmed down on it a bit since this article came out in 2015. I don't know what the latest fad food is, but I can almost guarantee that behind the fad is someone with money trying to make more of it.

Once a little-known speciality crop, its meteoric rise is now the subject of national news segments. Some experts are predicting that kale salads will soon be on the menus at TGI Friday's and McDonald's.

I don't know about Friday's, but apparently McD's did come out with a kale salad that had more calories than a Big Mac.

Calories (if you haven't been following along, I talked about calories a few days ago) aren't the only indicator of healthiness in a food, but I do find that hilarious.

This makes it pretty interesting that kale and cabbage — along with broccoli, Brussels sprouts, cauliflower, collard greens, and kohlrabi, and several other vegetables — all come from the exact same plant species: Brassica oleracea.

Linguistic detour, as I often do: you might notice that most of those have some variant of "cole" in them. It's in the middle of broccoli, the beginning of cauliflower and collard and kohlrabi, and it's a slight vowel shift for kale. Not so much cabbage itself, but the syllable survives in that version's most perfect presentation, coleslaw. The odd one out is Brussels sprouts. One might be forgiven for thinking that this is where the species name brassica came from, but apparently not; brassica seems to have been Pliny the Elder's name for the cabbage group. No, Brussels sprouts were named for, believe it or not, Brussels, the one in Belgium, near which the plant was heavily cultivated.

Nor was Brussels

apparently named after brassica. The similarity of the first syllables there seems to be another linguistic near-coincidence.

How is this possible? About 2500 years ago, B. oleracea was solely a wild plant that grew along the coast of Britain, France, and countries in the Mediterranean.

Because France doesn't count as "Mediterranean?" I guess they meant the north and/or west coasts of the country.

As for "how is this possible," obviously, the article goes into detail. But it's the same sort of thing as saying "How is it possible that a chihuahua and a Great Dane can be the same species?"

Short version: artificial selection; that is, trait selection by humans. As we, too, are part of nature, the distinction between artificial and natural selection is, well, an artificial one.

For the long version, there's the article.

All this speaks to one thing: the remarkable power of human breeding and artificial selection.

Wow, that could have been worded better, couldn't it? Oh, well; I'm sure I've done worse.

5 Like • Comment • View 2 comments

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