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 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.

January 25, 2023 at 12:01am

#1043629

Satan's Earwax

Yes, this has been languishing in my queue since October. The article itself is four years older than that, though.

Actually, Candy Corn Is Great
The reviled Halloween treat, which has deep roots in American history, should have a better rep

1) No, it's not. 2) No, it shouldn't. Candy corn is a vile abomination that could only have sprung from a warped, twisted, sadistic mind.

Much like the word “moist” and the music of Nickelback, candy corn is a thing that’s cool to hate. In an article titled “Candy Corn Is Garbage,” Deadspin points to “hobos, serial murderers, and Satan” as the only people who like candy corn; The Takeout, also driven to invoke the devil to describe candy in a candy corn debate, calls it “Satan’s earwax”; Buzzfeed, combining two pariahs in one pithy line, lists “the leftover crumbs stuck in Guy Fieri’s goatee” among things that taste better than candy corn.

While it's true that there are things that people love to hate due to bandwagoning, candy corn is not among those things. It's legitimately lame.

"Satan's earwax" cracks me up, though.

But here’s the thing: They’re all wrong.

"That's just, like, your opinion, man."

Candy corn, on the other hand, has been around since the 19th century, its roots firmly planted in American soil.

You know what else has roots firmly planted in American soil? Poison ivy.

What set candy corn apart was its revolutionary tri-color design: those white, yellow, and orange stripes. Done manually, by men pouring heavy buckets of steaming sugary liquid, the labor-intensive coloring process resulted in a visual excitement no other confection could match.

As the other candies around at the time were brown (butterscotch) or black (licorice), I can concede that point—for the time when it came out. These days, I doubt it's so labor-intensive, unless you're part of the Robot Union (local 3.14159), and... well, if you want colors, just look at Spree, Skittles, or M&Ms.

Today, the two major candy corn manufacturers — Jelly Belly and Brach’s Candy — use largely the same recipe Wunderle did back in the day (sugar and corn syrup, fondant, confectioner’s wax, and various other additions, like vanilla flavor or marshmallow creme).

Conveniently, this article glosses over the truth about "confectioner's wax," which is bug secretions.

Now, look. I admit I'm playing that for the ick factor. I mean, sure, it's real: there's bug goo coating candy corn. But honestly, that's not a problem for me. Consider that, first of all, lots of people eat insects. I've eaten insects, sometimes even on purpose. There's nothing inherently wrong with eating bugs. And, second, honey is also a bug secretion. Unless you're vegan, this shouldn't necessarily be a problem.

If I wanted to get technical, I'd point out that entomologists limit what insects they call "bugs," but for us normal people, "bug" can mean almost any insect. Just getting that out of the way so I don't get comments about it.

But no, my problem with candy corn isn't the insect content; it's everything about it.

The main difference is that the laborious hand-pouring process has been taken over by machines, which means that they can produce a lot of candy corn: According to the National Confectioners’ Association, American companies produce 35 million pounds, or 9 billion kernels, annually.

I told you they used machines. Rise up, my metallic brothers and sisters! You have nothing to lose but your chains!

But this prodigious production isn’t met with an equal amount of enthusiasm. A 2013 survey from the NCA showed that only 12 percent of Americans think of candy corn as their favorite treat (and they included “gum and mints” as an option, so the competition wasn’t exactly stiff).

Still, 12 percent is way too high, in my estimation, for the number of people for whom it's a "favorite."

With all the candy corn produced, and the apparent universal disdain for it, something doesn’t add up. One of two things is true: either people are lying about their candy corn opinions, or tons of candy corn gets thrown out each year.

I'm guessing both?

The notion that candy corn tastes bad is a lie. It’s just not true.

There exists a significant fraction of the human population for whom cilantro tastes like the devil's soap. It's a genetic thing. I'm not one of them, though I can't say I love it, either. But if I said "the notion that cilantro tastes bad is a lie," I'd get all kinds of rebukes.

Though the primary ingredient is sugar, candy corn’s flavor transcends cloying sweetness, becoming something richer and more nuanced: There’s a nuttiness reminiscent of marzipan, hints of warm vanilla, a buttery flavor belied by the fact that candy corn is, as bags proudly proclaim, a fat-free candy.

I don't exactly have the sharpest taste buds, but I do tend to taste nuance in things like beer, wine, scotch, and tequila. Candy corn, however, just tastes like sweet. No marzipan, no vanilla (a flavor I love), maybe a slight hint of butter? Not surprising there, because warm sugar tends to be buttery.

Being fat-free is a holdover from the fat-phobic 90s. Who cares if it's fat-free if it's nothing but simple carbohydrates? But we're not arguing about the health effects; it's candy, for fuck's sake.

This short texture resembles ear wax, or a candle (two common comparisons), only insofar as it has a slightly waxy exterior, created by the confectioner’s wax that gives candy corn its cheerful sheen.

Bug. Secretion.

But regardless, critics should beware the logical extension of dismissing a food because its texture resembles something else: Do we hate mochi because it has the texture of a rubber ball?

No matter how much I read, there's always a new food I've never heard of. What the hell is mochi? ...oh, a Japanese rice cake. Sometimes, I can be arsed to look something up. (Yes, I admire Japanese culture and love Japanese food; no, I haven't learned everything. This is a good thing.)

Do we revile yogurt because it’s the texture of body lotion?

No, I revile Greek yogurt because it's the texture of gooey chalk.

Do we recoil at flourless chocolate cake because it shares a texture with human waste?

Munched on a lot of shit, have you?

Leave your texture arguments at the door, please. They’re invalid.

Most of the people I know who dislike mushrooms have a problem with their texture. Texture is absolutely a part of the eating experience, and, as with taste, peoples' reactions are going to be different.

But I’m not here to denigrate other candies. Other candies are great! Reese’s Peanut Butter Cups are the greatest candy ever made...

No. No, they are not. The chocolate is waxy (can't be arsed to find out if that's from bug secretions or not), and the "peanut butter" is dry, vaguely peanut-flavored sugar.

I realize that RPBCs make it to the top of the list of "people's favorite candies" on an annual basis, so I know I'm swimming against the tide, here. I'm just pointing this out to show that I don't hold opinions just because they're popular.

Now, if someone made an RPBC knockoff, only more expensive, with dark chocolate and actual peanut butter, I'd become diabetic within minutes.

...Snickers truly do satisfy...

They're not that great. When it comes to chocolate/peanut combinations, though, I'll take a Snickers over a RPBC any day, even though I dislike peanuts but like peanut butter.

...and even tooth-destroying Butterfingers hold a unique place in my heart...

On a scale of one to "all the candy bars," Butterfingers are in the solid middle for me.

My love for candy corn doesn’t make me an antagonist to America’s most popular treats — and the assumption that it would is at the root of America’s abandonment of candy corn, and, dare I say, many other problems we face today: We seem to have forgotten that we can like one thing without hating another.

And finally—FINALLY—the author says something I can agree with. It's okay to like both Star Trek and Star Wars. It's okay to like both Marvel superheros and their DC counterparts. You could even like more than one sportsball team, if you really wanted to. It's not just this that I take issue with, but also the need to dump everything into "awesome" and "sucks" drawers without considering, as I did with the Butterfinger bar above, that some things are just okay.

Now, I should probably point out that I know that this writer is making a point with her editorializing. I recognize it, because I do it myself from time to time. And I kind of see her point, in the general sense: that we should draw our own conclusions about something and not love, hate, or feel something in between about something, just because everyone around you does.

She's wrong about candy corn, of course. It's disgusting. But she's right about the overall point.

After all this ranting, you may be wondering what my favorite sweet treat is. And I can't really answer that. Even though I don't munch on sugar very much these days, I'll get tired of one and move on to another. It cycles. So I'll just say "Lindt 70% dark chocolate" and leave it at that.

So what's your favorite / most hated?

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