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:

Elycia Lee ☮

about 7 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 23 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

February 29, 2020 at 12:01am

#976594

D'accord!

Etymology is something of an interest of mine, combining history and language studies... and maybe a bit of mob psychology in the mix.

https://www.mentalfloss.com/article/50042/whats-real-origin-ok

What's the Real Origin of "OK"?

Yes, this is from 2013. So what? It's not like language is going out of style anytime soon.

By Arika Okrent

Oh, I see what you did there, OK-Rent.

"OK" is the all-purpose American expression that became an all-purpose English expression that became an all-purpose expression in dozens of other languages.

Still not as versatile as "fuck," but, OK.

There may be more stories about the origin of "OK" than there are uses for it

Just as intriguing for me as etymology is folk etymology; that is, the stories we tell ourselves about the origins of words or expressions. Some of these are just blatantly wrong, such as the idea that the monosyllabic vulgarity I mentioned above originally came from the acronym "For Unlawful Carnal Knowledge" (now immortalized as the title of a Van Halen album) or "Fornicating Under Consent of the King," each of which expressions contains entirely too much Latin for to be taken seriously - but people still believe one or the other.

Or, one time, I used the phrase "rule of thumb" around an angry woman, who proceeded to chide me for using an expression that "came from an ancient rule about only being able to beat one's wife with a stick no bigger than your thumb." Which is patent nonsense, but OK. No point in arguing with people like that; they're not going to change their minds.

Point is, even the false etymologies tell us something about human nature and what we're inclined to believe.

The truth about OK, as Allan Metcalf, the author of OK: The Improbable Story of America's Greatest Word, puts it, is that it was "born as a lame joke perpetrated by a newspaper editor in 1839."

Lest anyone still be laboring under the misapprehension that comedy isn't meaningful.

It wasn't as strange as it might seem for the author to coin OK as an abbreviation for "all correct." There was a fashion then for playful abbreviations like i.s.b.d (it shall be done), r.t.b.s (remains to be seen), and s.p. (small potatoes). They were the early ancestors of OMG, LOL, and tl;dr.

And we thought the military went overboard with its acronyms.

OK got lucky by hitting the contentious presidential election jackpot. During the 1840 election the "oll korrect" OK merged with Martin van Buren's nickname, Old Kinderhook, when some van Buren supporters formed the O.K. Club.

This process, of a thought taking on a life of its own, is the sort of thing that Dawkins meant when he coined the word "meme," not captioned cat photos. Ironically, such a shift in definition is itself the hallmark of a meme.

Now, you might ask yourself, "How do we know this is true and not the other origin stories?" But that strays perilously close to "How do we know anything at all?" I find epistemology to be a fine, upstanding word, but not something I'd like to argue about.

But, as Metcalf says, its ultimate success may have depended on "the almost universal amnesia about the true origins of OK that took place early in the twentieth century.

And that's the fascinating part to me - to trace how these things develop over time. Sometimes we don't know and have to guess, but even the guesses can shed light on meaning. I'm OK with that.

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