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:

Write_Mikey_Write!

about 20 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 22 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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