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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10 Most Recent Blog Comments:

Writer_Mike

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

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

a day 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.

1 Like

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

1 Like

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

1 Like

Elisa: Middle Aged Stik

Sep 21, 2024 at 7:34am
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

Sep 21, 2024 at 6:46am
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

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
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.

January 20, 2023 at 12:01am

#1043396

'Sup, Y'all?

I thought y'all would want to see this.

‘Y’all,’ that most Southern of Southernisms, is going mainstream – and it’s about time

I use "y'all" as a second person plural, a part of speech that English otherwise lacks. And it can't always be inferred from context.

Southern Living magazine once described “y’all” as “the quintessential Southern pronoun.” It’s as iconically Southern as sweet tea and grits.

I like grits, but sweet tea can kiss my... ass.

“Y’all” fills that second person plural slot – as does “you guys,” “youse,” “you-uns” and a few others.

"You guys" is considered sexist these days, "youse" is still pretty much limited to a small area in the Northeast, and I'm not sure about "you-uns." I think Pittsburgh uses "yinz."

I’m interested in “y’all” because I was born in North Carolina and grew up saying it. I still do, probably a couple dozen times a day, usually without intention or even awareness.

I use it too, but more intentionally. I thought I used it more, but a quick search of this blog of over 2,000 entries only yielded 64 entries with "y'all." This would be #65.

Back in 1886, The New York Times ran a piece titled “Odd Southernisms” that described “y’all” as “one of the most ridiculous of all the Southernisms.”

Damyankees.

Like the Southern dialect in general, the use of “y’all” has often been seen as vulgar, low-class, uncultured and uneducated. As someone noted in Urban Dictionary, “Whoever uses [y’all] sounds like a hillbilly redneck.”

The only way to change this perception is to use it with intention.

The etymology of “y’all” is murky.

So is the etymology of a lot of other words.

My examples push “y’all” back 225 years before the citation in the “Oxford English Dictionary,” and they show that the word appeared first in England rather than the United States.

I think it’s important to point out that it originated in a more formal context than what’s commonly assumed. There are none of the class or cultural connotations of the later American examples.

Now, I can't be arsed to research this right now, but I think older versions of English made a distinction between second person singular and plural. That's how we got "thee" and "thou" and other constructions that are now associated with the KJV and maybe Shakespeare. Or something like that; like I said, not looking it up now.

Still, there it is, in an English poem written in 1631.

Not long after Shakespeare, really. Y'all Brits invented the language; we just perfected it.

“Y’all means all” – that’s a wonderful phrase that seems to be popping up everywhere, from T-shirts and book titles to memes and music.

Sounds good to me.

Now, how about we come up with a first-person plural that distinguishes between "us, including you," and "us, not including you?" Like if I said, "We're going to a party," does that mean you're invited? No. No, it does not, and now I'm embarrassed because you inferred that it did.

5 Like • Comment • View 3 comments

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