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 14 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 16 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.

April 23, 2020 at 12:22am

#981796

Disasters

Remember a few days ago, I said I'd get to another David Wong article from Cracked?

Today is that day.

https://www.cracked.com/blog/5-common-beliefs-that-make-disasters-worse/

5 Common Beliefs That Make Disasters Worse

As a keen observer of humanity and systemic failure, my pandemic lockdown has made one thing clear: Video game item shops should not close at night.

This is true. However, at least in the video games I play, it's easy to "wait" for several hours for the shop to reopen. This has no effect on the game other than making time pass in the game faster than it does in "real" life.

But the second thing that I've kind of noticed while waiting for the next Animal Crossing item balloon to appear is that disasters have a way of exposing all of our collective brain flaws.

Some asshole wrote an article condemning Animal Crossing recently. I won't link it because I don't want to give it any more publicity than it has already garnered. His point, if you can call it that, was that it's a childish game. I say, so what? You like to play it, then play it. I'm not a fan, myself, but then, how many people are on their 104th run-through of Skyrim? Besides me. We'd do a whole lot better without game-shaming. Or shaming of any kind. I suspect the article was written entirely to generate publicity, because the only thing more certain than trolls on the internet is people feeding said trolls.

This isn't because the world is trying to teach us a lesson -- the world is trying to murder us, not teach us -- but because information sometimes doesn't sink into the human brain unless it comes tied to a rock hurled through our window.

That in itself is an important lesson.

So, on to the Cracked-format numbered list.

5. "Freedom Means Doing The Opposite Of Whatever The Government Tells Me!"

My favorite thing about Americans is that we hate being told what to do.

Which is why game-shaming articles are so ubiquitous. "This asshole is telling me not to play this game! I'll show him by buying the game and playing the shit out of it!" That sort of thing doesn't work on me. At least I don't think it does. There was the time when I started taking a medication that interacts poorly with grapefruit. I've never craved grapefruit in my entire life... until the moment I started taking the medication. Still, craving something is not the same as doing it, and I didn't do it.

Sure enough, the moment the experts urged people to stay home to stop COVID-19, lots of us defiantly flocked to crowded beaches. But who, exactly, were we defying? The government? The virus? Death itself?

I've seen people, even here on WDC, saying stuff like, "If I get it, I get it." And that's an opinion I can respect. You have things you value more than your life or health? That's fine. You do you, as the saying goes. I know I do; I have all sorts of risky behaviors, like drinking, for which I've done an internal, informal, cost/benefit analysis. Well and good. But as Wong points out, it's not about you. It's about the people you could pass the disease on to. I say - controversially - go ahead and play Russian Roulette. It's your skull. But the moment you point that revolver at me, or anyone else, we're going to have a Problem.

Hmm, it's almost as if our society has built up selfish dicks as a heroic archetype...

4. "In A Disaster, It's The Tough Loner Who Survives!"

Some of us literally prefer the fantasy of mass death to the reality, which is that workforce specialization has turned the entire concept of rugged independence into a selfish, childish daydream. Maybe you can learn to grow your own food and purify water, but you sure as hell can't manufacture vaccines or perform a root canal on yourself. Side note: I'm convinced you could kill the post-apocalypse genre forever just by attaching a device that lets the audience to smell the characters. Though I guess the fantasy genre would die with it.

Reality: We're interdependent. Thoreau wasn't just wrong; he was dangerously wrong. In times like these, or with other natural disasters, what's important is how well we treat each other, not who has the most rolls of asswipe.

3. "If I Prepare For A Disaster And It Doesn't Occur, Then The Preparations Were A Waste!"

But the real shock of the Corona Crash has been finding out that apparently our entire infrastructure was operating on this same, "only plan for what you need five minutes from now" philosophy. The US government's emergency pandemic stockpiles were tiny and withered. The billionaires who scolded the poor for not having six months of emergency cash immediately screamed for government bailouts after a single week of bad sales.

I've harped on this sort of thing before. I've noticed that no one is talking about Marie Kondo or that Hoarders show right now. I don't know; maybe it's because the hoarders had the right idea all along? Okay, some of the more extreme ones are just mental, I know; your stash of used pizza boxes isn't going to do you a damn bit of good right now unless you become homeless and have to create a shelter out of them.

If you cut back to plan for the future, they say you're not "Living in the moment." If you stock up on emergency supplies you're a paranoid prepper, if you wash your hands too much you're a germaphobe, if you worry about potential disasters then you need to "Stop and smell the roses." You know, for your own mental health.

In fact, if tomorrow we all collectively decide to slow down, simplify our lives and save for a rainy day, I can tell you exactly what the media will call it: A Worldwide Economic Collapse.

2. "If Someone Warns Of A Disaster And it Doesn't Occur, It Means They Were Wrong And We Should All Laugh At Them!"

If you save a drowning child in a lake, you'll get your face on the local news. If you put up a sign and a fence that keeps all children out of the dangerous lake entirely, you're just the cranky jerk ruining everybody's fun. "Look, it's the mean 'no swimming' guy! Eat my dry ass, old man!"

I'm skipping a bunch here, but they're all valid points.

1. "Handling This Is Surely Someone Else's Job!"

It's just basic game theory: why should you make the sacrifice if there's a chance someone else can do it instead? If you don't buy up all of the toilet paper, some other hoarder probably will. Why should they be the one who gets to spend quarantine doing their wacky Mummy character on Tiktok?

I've said this before, too: hoarding toilet paper? Fine. But it's insulting to the rest of us when you take your seemingly-limitless supply of bumwad and do Stupid Human Tricks with it instead of wiping your ass with it. It's like those douchenuggets who light cigars with $100 bills.

Incidentally, I used to wonder why people lit cigars with anything other than a match, until I started smoking cigars. Turns out that the sulfur in a match can ruin the taste of a good cigar. So you light something else first - a cedar chip is preferable to a goddamned C-note, and about 10,000 times less expensive - and use that to light the stick.

In the end, our inability to prepare for the worst is really just A) an inability to grasp which hypothetical futures can actually occur and B) an unwillingness to see strangers as real humans whose needs overlap our own.

I've often wondered about the mentality of those who buy gold against the inevitable fall of civilization. First of all, it's gold. You can't eat it or drink it (well, I suppose you could, but the results range from nothing to burning your digestive tract), and if you build a shelter out of it, someone will come along and kill you for it. Come the actual apocalypse, gold won't be worth the paper it's printed on. Hell, you can't even build effective weapons out of it; it's too malleable. Sure, gold is an excellent conductor of electricity and has many nice industrial uses -- all of which are immaterial in the face of a return to the Stone Age. That's assuming you even survive, which you won't (neither will I).

And second, if gold were so useful, why are people selling it? What do they get out of it? Money. I have this sneaking suspicion that actual paper money will be more useful after a collapse, in spite of everyone's assumption that it'll be worthless. It's not like they'll be making more of it, and, unlike gold, it has a shelf life.

Besides, in a pinch, you can wipe your ass with it. Try doing that with a hunk of yellow metal.

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