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 8 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 10 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.

June 24, 2021 at 12:02am

#1012432

No Accounting for Taste

One of the reasons I like to travel -- I'm still planning on leaving on Saturday -- is to eat at restaurants I haven't been to.

6 Everyday Foods That Only Became Popular Relatively Recently

Don't get me wrong, though; I'm not one of those food adventurers. I'll leave the bug-chomping, weed-chewing, and snail-juice-drinking to much bigger attention whores. Why try anything exotic when, for example, I still haven't found the Perfect Cheeseburger? (I did find the Perfect Pizza once, but that's not going to stop me from trying other places.)

But every common food was new once. I'm pretty sure I posted in here before about tomatoes -- how Europeans, once those red round bastards made it across the ocean, at first thought they were poisonous. Now you can't spit in Europe without hitting a tomato-based dish. And who in the hell first looked at a cow and went, "I'm just going to squeeze these dangly things here and drink whatever comes out?"

Point is, today's staple is yesterday's adventure food, so I can't rag too much on the brave explorers of culinary experimentation. It's just not for me.

So today's article, from Cracked (and as usual, selected at random), is about foods that are new. Ish.

Have you ever completely changed your opinion on a food? Maybe you had PB&Js all the time as a kid, then one day, decided it wasn't for you.

Nope. Still like 'em. Though I can never be arsed to have all the ingredients for them, and they're not exactly a staple at restaurants.

Maybe you hated canned tuna, but then one day ordered ahi nigiri to seem sophisticated on a date and now you own a tuna fishing boat.

Nope. Still like tuna. Yes, I heard about Subway. All I'll say on that is: absence of evidence is not evidence of absence.

Maybe you thought New York pizza was the pinnacle of the universe, and then you spent a weekend in Chicago.

You shut your whore mouth.

6. You're Not Turning Into Your Parents, Brussels Sprouts Actually Do Taste Better Now

You know how a big gag in the '80s and '90s was about how terrible Brussels sprouts were?

Um... no? I mean, yes, I hated them as a kid because my mother cooked them and, while she tried, she couldn't cook vegetables to save her life. Which sucked because we had an enormous vegetable garden.

One day, I'm checking out a big bag of Brussels sprouts at the grocery store, because once I learned how they were supposed to be cooked (that is, not boiled for an hour and kept unseasoned), I started liking them.

Cashier: Can I ask you a question?

Me: Sure.

Cashier: How can you eat those things?

Apparently, she never got the memo.

Anyway...

Hans Van Doorn, a scientist with a name so Dutch we don't have to tell you where he's from, isolated all the compounds that make Brussels sprouts so bitter. Then, a handful of Brussels sprouts companies (apparently there are whole companies devoted to this stuff) started looking at records of old varieties with lower amounts of these compounds. One breeder at the Bejo Zaden company said they had "a whole gene bank here in our cellars."

How it's possible to trust a comic-book villain is still unresolved.

Apparently that gene bank is organized into "this sucked" and "this didn't suck," and the breeders got their Gregor Mendel on and started experimenting with various groups, cross-pollinating and cross-pollinating over and over until people didn't hate Brussels sprouts quite as much.

Science!

Also, it should be noted that Brussels sprouts and the next entry, kale, are actually the same plant. Along with cabbage, turnips, and a whole list of other things your mom made you eat before you could have dessert. (This is somewhat similar to how chihuahuas and great danes are both dogs.)

5. Kale Went From Decoration To Ubiquitous Fad

Kale: just typing the word probably made you reflexively click over to a Pinterest board.

Nope.

We grew kale, too. Even managed to sell some at farmer's markets. This was back in the 70s. We'd have charged more if we knew it was going to become a joke.

Pizza Hut relegated kale to ... salad bar decoration. No, Pizza Hut didn't even have some secret soul food era where they slow-cooked greens with smoked ham hocks. They just bought bunches and bunches of kale because it looked healthy on the salad bar.

It is an invariable truth that anything that "looks healthy" is going to taste like warm ass. Kale is no exception.

Speaking of trendy foods...

4. Avocado Became Popular So Fast Cartels Got Involved

No, we didn't grow avocados. Wrong climate. But when we got them at the store, I hated them because my mom's idea of a ripe avocado was one that was mostly black inside. In her defense, there's an approximately fifteen-second window between "underripe" and "rotten," and it's really hard to hit that window just from looking at their wrinkly outsides.

If you're an American over the age of 30, you probably remember a time when avocados weren't absolutely everywhere (if you're an American under 30, please keep reading, but don't tell me).

No, not really. Like I said, I remember avocados from way back. I even like them. I simply don't worship them. I can take or leave guacamole, but sometimes they're a nice accompaniment to beef.

That's because avocados mostly grow in Mexico -- tropical fruit doesn't exactly thrive in, say, Indiana. To be fair, nothing thrives in Indiana, but avocados have an especially hard time.

Actually, I just quoted the above bit to reinforce that it's true that nothing thrives in Indiana.

3. It Took Less Than One Generation To Completely Change How Babies Eat

On the list I made long ago, "Pros and Cons of Becoming a Father," there was, among dozens of other entries, a triple-underlined phrase on the "Con" side: "Baby food is gross."

I don't remember the one or two things that were on the Pro side.

But purees are slowly becoming as popular as pregnant women having three martini lunches. In its place is something called Baby-Led Weaning, which is when parents set pieces of solid food in front of their baby and let them pick around and explore tastes.

You mean "throw some of the food against the wall, scatter most of the rest on the table, and discover the immutability of gravity by dumping what's left on the floor."

Also, recycling those endless Gerber jars sounds like a nightmare.

My dad had endless empty Gerber jars in the basement. They were excellent for storing screws. He used them right up until I had to move him due to Alzheimer's, when I was in my 30s.

2. The Plant-Based Burger Craze Is Kind Of Miraculous

It wasn't so long ago that vegetarian burgers were "black beans pressed into a patty" or "a weird mush of a bunch of veggies, with corn conspicuously sticking out."

I like meat. As I noted above, I'm on a quest to find the Perfect Cheeseburger. But I've also been known to enjoy plant-based substitutes for meat, earning me the scorn of some fellow carnivores. I don't care; I like what i like. Traditional veggie burgers are good, in my opinion, not because they taste like burgers (they totally don't) but because they're their own thing. Veggie "sausages" are actually quite good. And vegetarian "bacon," while it doesn't taste even the slightest bit like actual bacon, isn't bad; I just don't try to pretend it is bacon.

Vegetarian "chicken nuggets," on the other hand, taste like cardboard.

Anyway, the point is, I'm not above trying vegetarian food (though I will generally shun anything labeled "vegan" on principle). I haven't had occasion to attempt these newer, supposedly burger-tasting burgers, but I will. I just want to try the one that doesn't contain beet juice. Beets taste like cardboard soaked in a mud puddle.

Some point to these companies' strategy of trying to appeal to carnivores as the key to their success. This makes sense -- being told you're getting chicken tenders when all you're really getting is pressed tofu sucks -- but you can't always rely on red-blooded American burger-chompers to completely change their habits because you did a science. If the response to the COVID-19 pandemic says anything, it's that Americans do not give one single 7-11 hot dog's worth of poop about science.

I'm sure that helps. It would help more if they were cheaper. After all, if a thing is either a) made out of something that eats plants or b) made out of plants, it makes a lot of sense for the one that skips the middlemoo to be cheaper. Alas, they are not.

1. Centuries-Old Rich Snobs Are Why European Food Is Spice-less

You know how white people can't handle spice? Shut up, yes you do. I don't care if you're outraged because you're white. I'm white.

Sod off. I feel insulted every time I walk into a Mexican or East Asian restaurant and it tastes like they've toned down the spices because I have light hair, blue eyes, and vampire-pale skin. Don't make assumptions; that's racist. See that ghost pepper over there? Gimme.

Cinnamon wasn't always on your cereal and paprika wasn't always on your Doritos. Those spices come from somewhere, and that somewhere is (again) not Indiana.

If it seems like I take great pleasure in insulting Indiana, it's because I do. I was born there but had the good sense to escape at the ripe old age of three days. In a few days, I'll be passing through again. Quickly.

The rest of this item I won't paste, but go read it; it's about how colonialism ruined everything, including the taste of food.

So anyway, now I'm all hungry. Still not going to eat bugs, though.

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