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 3 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 5 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.

1 Like

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').

1 Like

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.

September 26, 2020 at 12:08am

#994223

O'zapft is!

Hey, this one's pretty easy for me.

PROMPT September 26th

It’s apple-picking season where I live and I’m so excited! Write about your favorite seasonal activities. What is something that only comes once a year, but you always look forward to?

I'm not a fan of Christmas trappings, and New Year's just brings fresh horrors. Thanksgiving does little for me, because my family is mostly gone and I won't cook a turkey for just myself. Halloween is kinda fun with all the spooky stuff, but it's not really a big deal for me, either. My birthday would be kinda cool, but it's in February, and I hate February. Easter isn't something I celebrate, and Pesach means schlepping to my few remaining family in New York City (not this year, though) and trying to find a parking spot in Manhattan on a holiday weekend. You'd think that being a would-be comedy writer, April Fools' Day would be my holiday, but I tend to spend it in my room under the covers, lest someone mistakenly assume that it would be funny to play a prank on me (it would not). Memorial Day, Independence Day, Labor day -- all just another day for me because I'm retired.

But there's one thing I look forward to every year, and -- you guessed it -- it involves beer.

I speak, of course, of Oktoberfest (which is really celebrated in late September because Germans are weird). (I can say that because my dad's side of the family was German.)

So it's going on right now, and the one thing I look forward to every year is getting to drink Oktoberfest beer, aka festbier, aka Märzen lager.

The story behind why Oktoberfest beer is named after the month of March is an interesting one, but I can't be arsed to look it up right now. As I recall, there was a period when Germany, or rather the nation-states that would later become Germany, forbade the brewing of beer between, like, April and August, or thereabouts. This had to do with refrigerators having not been invented yet. So they'd brew some beer in March and let it sit in barrels in caves or some shit until it was time for Oktoberfest (which itself started out as a wedding celebration for royalty).

Like I said, not looking it up, so I might have some of the details wrong, but that's the gist of it.

Anyway, point is, these days Märzen lager starts hitting shelves around late August, and I drink the hell out of that shit. My favorite is a local one, an Imperial version called Hansel and Kettle that's high-octane and delicious. The "Imperial" style name is borrowed from Russian Imperial Stout, which isn't actually Russian but British, and that's a whole 'nother story that I won't get into right now. It mostly just indicates that a beer has a higher ABV than usual, and that other ingredients have been balanced out to compensate.

But I'll try any festbier, if I'm at a taphouse or whatever, and they usually have a consistent quality to them because the recipe doesn't allow for too many variations.

Another thing that lights up what's otherwise a crappy season for me -- I know a lot of people like fall, but I despise cooler weather and prefer the sweltering heat of summer and, especially, its mild nights here -- is pumpkin beer.

But pumpkin beer varies wildly in quality. Some don't even use much pumpkin, just the ever-present fall staple of pumpkin spice. Some are way to sweet for my taste. Others are entirely too much pumpkin, not enough spice. But when I get a good one (Pumking from Southern Tier, out of New York State, another Imperial variant, is my favorite), I drink it. Still, it could go away and it wouldn't affect me much, as long as I get my Oktoberfest.

I have a vague desire to go to actual Germany for the celebration sometime, but it's usually really crowded there, and I don't do well with crowds, beer or no beer. Still, for the experience, it might be worth it.

So that's the big thing I look forward to. As the weather continues to turn against me, and Märzen becomes but a memory, I'll probably be drinking more Russian Imperial Stout. But that's not seasonal. Festbier is.

I'm told that the title of this entry translates to "It is tapped!" which traditionally opens the festivities in Munich. But I don't know much German, so hopefully it's not actually something rude.

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