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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WakeUpAndLive️~Happiness

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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
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WakeUpAndLive️~Happiness

a day ago
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WakeUpAndLive️~Happiness

a day ago
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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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October 31, 2023 at 10:59am

#1058353

Punctuated

It's a 2.5-year-old article from LitHub, but it's not like style ever goes out of style, right?

The Punctuation Marks Loved (and Hated) by Famous Writers
; vs. — vs. , vs. . vs. !

Parul Sehgal once argued that style “is 90 percent punctuation.”

Sure, not like anyone grumps about word choice.

As John Mayer once apparently said, for some reason, “Ladies, if you want to know the way to my heart: good spelling and good grammar, good punctuation, capitalize only where you are supposed to capitalize, it’s done.”

Apparently, dudes could get away with anything.

After all, even among experts, there are disagreements, some of them oddly vehement. (What inner forces would compel someone to demonize or deify the semicolon?)

Oh, I don't know... search my blog for "semicolon" and you might find out. (I'm a fan of it, obviously.)

Then the article goes into the punctuation marks in general. I'll only highlight the ones I agree with:

(semicolon) In compiling the sentence, efficacy—or, more precisely, precision—is important; capacity is important; and clarity is important. This kind of writer, at least, doesn’t think in little stoppered declarative sentences. It isn’t like that. Not really ever. Perhaps for some people. But not for us. For those of us whose thoughts digress; for whom unexpected juxtapositions are exhilarating rather than tiresome; who aim, if always inadequately, to convey life’s experience in some semblance of its complexity—for such writers, the semi-colon is invaluable. -Claire Messud

I have never heard of this author before, but that makes me want to seek her work out.

(exclamation point) Keep your exclamation points under control. You are allowed no more than two or three per 100,000 words of prose. -Elmore Leonard

I think this is one of those things that's going to be different depending on what you're writing. But yeah, I believe in using bangs (easier to say than "exclamation point") sparingly so as to maintain their power. Unlike cuss words, which, as my mom's family was from New York City and my dad was a sailor, to me, are really just punctuation marks.

(em-dash) Don’t you find it annoying—and you can tell me if you do, I won’t be hurt—when a writer inserts a thought into the midst of another one that’s not yet complete? Strunk and White—who must always be mentioned in articles such as this one—counsel against overusing the dash as well: “Use a dash only when a more common mark of punctuation seems inadequate.” - "many editors"

Contrary to what I said above, I don't really agree with this, but as with the bang, sparingly is good. I can see right through this writer's ham-handed attempt to convert people to their point of view by deliberately overusing it.

(comma) [On her editor, Bob Gottlieb, who famously “was always inserting commas into Morrison’s sentences and she was always taking them out”] We read the same way. We think the same way. He is overwhelmingly aggressive about commas and all sorts of things. He does not understand that commas are for pauses and breath. He thinks commas are for grammatical things. We have come to an understanding, but it is still a fight. -Toni Morrison

Oscar Wilde rather famously once wrote: "In the morning I took out a comma, but on mature reflection, I put it back again." As the most-used punctuation mark apart from the period ("full stop" for those of you across the pond), it's subject to a great deal of wasted words and time. I admit to feeling somewhat judgemental toward people who misuse them, but I can't always articulate why they're misused in a given instance.

I'm more judgemental about apostrophe abuse.

(hyphen)

No quote here. I did a whole blog entry on it a while back: "Dashing"

. In it, I overused and misused emdashes, so take it as you will.

(period)

Do we really have to argue about full stops?

James Joyce is a good model for punctuation. -Cormac McCarthy

James Joyce is not a good model for anything, and now I have even less desire to read anything McCarthy wrote.

I mean, if you write properly you shouldn’t have to punctuate. -McCarthy again

And that sentence, folks, should be in the dictionary as a sample under "irony."

To be clear, I'm not trying to claim I'm always right. I'm aware that I overuse semicolons sometimes (as well as parentheses), and get things wrong on occasion. Especially here, where every entry is a first or second draft. But that doesn't mean I don't consider these things.

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