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 16, 2020 at 12:20am

#993461

Pappy? Happy.

This is going to be a tough one. I don't do "happy."

PROMPT September 16th

What is your happiest memory? Describe it in vivid detail!

It should surprise no one that all of my happy memories involve drinking.

Oh, I'm sure that there were some from my childhood floating around in there somewhere, but that was so long ago that the memories can't be trusted.

So I'll tell you about the second time I managed to drink a shot of Pappy Van Winkle.

Pappy is a bourbon. But to simply call it bourbon is like calling Eta Carinae a star.

...that's not very helpful, is it? If you're going to use a simile, it should be one that most people are familiar with. "A wolf is like a dog" can work, but "E. coli is similar to salmonella" probably doesn't help a reader very much at all. Let's try this again.

Pappy is a bourbon the way the Bikini Atoll hydrogen bomb was an explosion.

It's not easy to find. Rumor has it that small quantities appear in the liquor stores around here at whatever time they get their deliveries, and it sells out within seconds. I've never managed to show up at just the right time. It's also, as you might imagine, not cheap.

The first time I tried it, I found it in -- and keep in mind here that like most (but certainly not all) bourbon, it's made in Kentucky, a state that borders my own -- a bar in Livermore, a town in central California. I tried it, and the sky opened up and an angelic choir sang Hallelujah. The Handel one, not the Cohen one. That's a good memory.

But a better memory for me is actually the second time I tried it, because when drinking, it's not always about the beverage itself, but the other things going on in time and space proximity. To clarify, that first time, in Livermore, I'd already been having a good day, so it was just another great thing to happen on a great day.

The second time, I was staying at the Ritz-Carlton in Washington, DC. I've mentioned that visit before. I was attending a conference -- the details of which are entirely irrelevant -- and that evening, an optional part of the agenda was to go see a Nationals game.

I'm not a fan of baseball; I'm a fan of beer, so that evening I did what I usually do when traveling, which is to visit local breweries and sample their wares. Three of them were kind of within walking distance, so that's what I did.

Now, understand that while I like beer, part of the process for me involves trying beer I don't like. This provides contrast and context. As a great philosopher once said, "If everything was cool, and nothing sucked, how would we know what was cool?" Still, usually, on these visits, I can find one or two beers that I really like, a bunch that are okay, and a couple of dogs.

These breweries were a barking kennel.

By the time I was done, the sun was setting, so, to avoid walking past the White House after dark (bad neighborhood, you know), I took an Uber back to the hotel. Once I got there, I sat down at the bar. After that disappointment, I needed a positive drinking experience.

"What can I get you?" These timeless words, uttered by bartenders everywhere (at least ones who speak English), are like music to me. I started to feel better already.

I looked over the fancy drinks menu. Pricey, as you'd expect, but not too out of line considering it's the goddamn Ritz-Carlton. I settled on an Islay malt scotch. Smoky, peaty, reminiscent of fine leather, oak, and perfectly toasted bread.

Whilst savoring this sweet nectar of the Gaelic gods, I struck up a conversation with a nice lady who happened to be sitting at the bar near me. No, it wasn't like that. I'm done with that crap. Just another whisky drinker to talk to. But old habits die hard, and lizard brain went, "pRetTy lAdy aT bAr. mUst imPreSs prEttY laDy." So I ordered another scotch, this one a wee bit pricier.

When the bartender served that to me, I noted, "You know, I was expecting a place like the Ritz-Carlton to have Pappy Van Winkle."

Bartender goes, "Oh, we do. It's in the safe."

Yes. This bar has a safe to hold all their really good stuff. Lady and I followed him as he opened an honest-to-gods metal safe with one of those spoked wheels like in the movies.

Opening it was like opening the Ark of the Covenant in Indiana Jones, but not from the Nazis' point of view. No face-melting, that is, just bright light streaming out, backlighting a collection of green and amber bottles. I blinked, taking in the beautiful sight.

The price list, located on the inside of the open door to the safe, was where my gaze went next. Already regretting that annoying primal urge to show off, I steeled myself to be shocked at the numbers.

But you know? They weren't all that bad. I've seen more expensive collections in Vegas. I guess DC can't count on people celebrating their poker winnings by buying the good stuff.

The only thing that surprised me was that Pappy was not the most expensive nectar listed. Fortunately, Pappy was the one I'd mentioned, so that was the one I felt obligated to purchase.

And I did, and savored that couple of precious ounces of golden dew over the next half hour, continuing to talk to whatever her name was (she was waiting for her husband to come back from the baseball game, another reason why I was only interested in conversation).

The point is, I guess, that the reason I'm calling this my happiest memory -- ask me tomorrow and I'll probably come up with a different one -- is the contrast. From drinking decidedly mediocre beer and walking around muggy, hazy DC in the summer heat, wiping sweat from my face at every third step, to sitting in an air-conditioned bar sipping some of the best whisky in the world, unexpectedly, and having a pleasant chat with a fellow drinker of the good stuff.

So yeah, that was a good day: avoiding baseball, discovering beers I didn't like, and then capping it all off with the good stuff. And forever being able to brag about having stayed, once, at the Ritz-Carlton.

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