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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 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 4 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 6, 2020 at 12:31am

#992517

I Drink and I Know Things

Holy shit, it's Sunday already?

PROMPT September 6th

Reflect on the last week. Write about something you did really well last week and something you could have done better.

Warning: I'm going to wallow in self-pity here. Skip this one if you're just going to roll your eyes and tell me to get over it already. I know everyone else has it worse than me. I know I should count myself lucky. I know I have no "right" to complain, but goddammit, this is my blog and I'll fucking complain if I want to.

The answer to both sections of the prompt is "drinking."

In the Before Time, I talked about drinking way more than I actually drank. Hell, I managed to drink so little in 2019 that, in conjunction with changing my eating and exercise habits, I lost 100 pounds. But shit, it's a big part of my brand, so I had to talk about it anyway. I make that sound so corporate, I know, but I really do enjoy the experience - just not enough to have made it a daily thing. Occasionally, I'd go to a brewery and sample a small amount of several of their beers. Once in a while, I'd pour myself a scotch, or mix a margarita, or go to a wine tasting. Then I'd go days, sometimes weeks, without touching a drop (I do have an unmitigated addiction to Coke Zero, but that hardly counts).

And once in a blue moon (callback to a previous blog entry here), I'd get drunk and pass out. Sometimes I just needed that. I don't get violent or nasty or neglectful when I do that. Usually I just listen to Brandi

Carlile or Leonard Cohen to enhance the emotion, and then pass out. And for the gods' sake, don't give me shit about driving drunk, because I never do that. Somehow we've conflated drinking and driving as a society, perhaps because everyone drives everywhere, but even when I'm completely cabbaged, I have enough sense to summon an Uber if I'm out -- but usually I do this at home.

My... temperance, if you want to call it that; the term originally referred to moderation, not teetotaling -- it wasn't a hardship for me; my other great love in life is video games, and I can't do those if I'm wasted. It requires coordination, thought, and concentration, none of which is compatible with being drunk.

I also like to drive. As I've mentioned before, I've driven myself across the goddamn continent three or four times now -- yes, I've lost count. And again, I'm very, very careful about being sober when I drive. Not for my own sake, but for everyone else's; same reason I wear a face mask in public these days.

Which brings me to how things have changed this year.

The thing I most love to do when I travel is visit breweries. I wish I'd kept better track, but I've had beers from close to a thousand craft breweries all over the country. As I said, I'm careful about it; I've been known to choose lodging based on it being in stumbling distance of a brewpub -- much to the detriment of the quality of said lodging. Thing is, I wanted to -- I still want to -- visit every brewery in the US and Canada. As of January 1, there were nearly 10,000 of them in my country alone; I suspect that number is substantially lower now, but I haven't looked into it because the last thing I need right now is to get more depressed.

I'm well aware that this is probably an unattainable goal. Every week in the Before Time, on average, we gained two and lost one. By the time I've swept the country, a hundred more will have popped up and I'd have to go back. But everyone needs a goal, and this one is mine.

Then the pandemic happened, and my world shrank to a pinpoint.

Now, as I said, I'm aware that I'm a fortunate, privileged individual. I have time because I'm retired. I'm financially stable; though far from filthy rich, I'm not exactly dirt poor (I always had it in my head that I'd write a personal finance guide titled "Dirt Rich" as a play on these clichés; can't be arsed to actually do it because I don't really have the credentials for it.) Most fortunate of all, I live within walking distance (just barely) of my second-favorite brewery in the US. (My first-favorite brewery is on the other gods-be-damned coast.) And I have a good, reliable Subaru to take me where I want to go.

The last time I moved that car was six months ago. It's been sitting, neglected, on the street in front of my house.

Which is not to say I haven't gone anywhere. I walk to the local taphouse. I walk to the aforementioned second-favorite brewery. I had a dentist appointment on the other side of the city a couple of weeks ago, for a toothache that I had developed precisely 15 seconds after my state shut everything down, and I took an Uber -- because the dentist office was near a brewery I hadn't been to, and I wanted to sample their wares.

Doing that, though, just made me more depressed. I got my usual sampler, just small portions of several beers, and they were good enough, but all it did was remind me of the thousand or so times I'd done the same thing in New York, New Jersey, Maine, Massachusetts, California, Nevada, Illinois, Texas, North Carolina, Iowa, Colorado, Florida, motherhumping Hawai'i... every state in the US, in fact, except for Michigan, Alaska, and Nebraska.

Technically, sure, I can travel. In practice, it's a bad idea.

So... getting back to the point here, finally... I drink. A lot. Every day. Well, almost every day; I skipped last Sunday so I could finally get some decent sleep.

Couple of weeks ago, the movie theater reopened, with precautions. More importantly, the taphouse in the movie theater reopened. By "the movie theater," I mean the Alamo that's a mile from my house. I walked there and staggered back. The movies -- Unhinged, New Mutants, and Tenet -- all kinda sucked, but it didn't matter; it made me feel halfway normal again, even though I was the only one in the theater for all of them. Well, except for New Mutants, where a couple of douchebags had seats way up in front while I was in the back. Anyway, I could have seen gods-be-damned Manos: The Hands of Fate (without the MST3K commentary) and it wouldn't have mattered, because the important thing was I was able to get cooked in the taphouse first.

And I discovered that I have forgotten how to interact with people.

Not that I was ever very good at it, but, like everything else, it's a skill that fades away if it's not used. I mean, yes, I have a housemate and we talk occasionally, and another way I'm fortunate is that we're still friends even though we're in close proximity, but in the last six months I've seen one other friend once, and an old friend and her husband... also once. My interactions with people have been limited, is what I'm saying. And so I forget how to be polite to bartenders and such. It's important to be polite to bartenders. They're a lifeline. It's not that I didn't try; I just feel like I failed at being sociable. Don't worry; I ameliorated any bad feelings with enormous tips. I still feel bad about it. It wasn't anything in particular that I could explain; it's just I was probably not as affable as I usually am.

So of course that means I'm even more inclined to drink at home. I've been ordering wine online. Near the movie theater is a liquor store, and I've staggered home carrying scotch, gin, tequila, and ingredients for my creation, the American Election (it's a twist on the White Russian; an American thing with Russian influence, hence the name). My second-favorite brewery (see above) delivers beer. The grocery store delivers beer and wine. I'm set. Who needs to travel, anyway, right?

Now, I've heard that the first step is admitting that you have a problem. So okay, I admit I have a problem.

You know what?

I don't give a shit.

In fact, I like it.

After I've posted this I'm going to use the last of my vodka to make another American Election, and I'm going to watch the last two original series Trek episodes.

First, though: once again, I'll have a free MB to give away tomorrow, I think the last one for Birthday Week, and it'll go to someone who makes a comment here below that I like. And yeah, if you've gotten one from me already this week, feel free to comment anyway, but the point is to give away the free one so it's not like I can wait two weeks for CR credit. Remember, though, I do this about once a week or so anyway, so there will be other chances. My account anniversary is coming up soon, and I'm sure I'll give out presents for that.

Today's MB goes to *drumroll* Turkey DrumStik

, because her comment about omniscience reminded me that it's one superpower that I definitely don't want. I mean, yeah, I love to know things... but there are some things about which even I prefer to remain ignorant.

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