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 ✍️

less than a minute 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.

Annette in France

about 11 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

about 15 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 15 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 20 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 21 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 21 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

May 5, 2024 at 7:04am

#1070539

Revisited: "Fail"

We're going back in time, as usual for Sunday mornings. This time, it's just to October of 2020 (specifically, 10/10/20, one of the few dates that work in both American and European style conventions and which can also be turned into a mathematical equation): "Fail"

I did the entry for that month's round of "Journalistic Intentions"

[18+], which is still an active blogging challenge, though not every month. Also, all the links in the entry are still solid, which isn't too surprising for something less than 4 years old.

Anyway, I got curious and looked up the origin of the name Fail.

I didn't go into this at the time, but today, I noticed that the ultimate origin of the Scottish surname Fail (which has many alternative spellings) is the name Paul. Don't ask me what kind of linguistic vowel shifts it takes to get from Paul to Fail; I don't know. But the name Paul, from Paulus, is Latin in origin, though it is probably most associated with the Paul of the New Testament.

So Paul becomes Phaill (or some similar spelling), and, by the convention of some Scottish clan names, generates MacPhaill, later Anglicized as MacFail or just Fail.

At least, that's one possible explanation. I never did trust some of these genealogy websites. But you know the Meat Loaf character in Fight Club? "His name was Robert Paulson." Well, "Paulson" is the English equivalent of MacPaul, which we've already speculated became the name Fail, so I have to wonder if Palahniuk did that shit on purpose. Probably not; his symbolism tends to be more of a, well, a punch in the face.

Anyway, back to my original entry:

I know, I know, it's rude to make fun of peoples' last names. You should be proud of your last name, even if it's Fail.

Or Hogg, as in yesterday's entry (coincidentally, also Scottish). Still, being proud of being a Hogg (or a Fail, for that matter) is no damn excuse to name your kid Ima.

There's the story of a Dr. A. Hedgeh, for example, who apparently had a problem with people adding "og" to his nameplate.

In case you don't want to bother rereading my 2020 entry, I'm including that link

here again, because I still laugh every time I see it. Fake or not.

So, yes, it's rude to make fun of peoples' names.

But sometimes, it's so funny you can't help it.

3 Like • Comment • View 1 comment

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