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.

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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').

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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.

February 10, 2022 at 12:02am

#1026383

Life is Like a Box of... Well, You Know.

A recent addition to my queue, and I'm already sharing it -- just in time for other people who are not me to prepare for Valentine's Day.

Why American Chocolate Tastes Like Garbage

In my town there is a chocolatier called Gearhart's. They make absolutely delicious chocolate, some of the best I've ever had. The headline, however, is referring to the bigger, mass-produced chocolate that does, indeed, taste like garbage.

In this way, chocolate is much like beer: the big guys have deluded people into thinking that the shit (or piss) they squeeze out is how it's supposed to taste, whereas nothing could be further from the truth.

Oh, and it's a Cracked link. I need to mention this because some people have reported browser issues on that site, which I don't experience because I have browser extensions that make my world much more peaceful.

In case you didn't know, the entire rest of the world mocks American chocolate as pukey, powdery, waxy garbage, and they're completely right to do so.

Yes. Yes, they are.

We're supposed to be the greatest country on earth—why can't we get something we love as much as chocolate right?

Not only that, but it comes from this continent. ("Greatest country on earth" still debatable.)

4 Blame Hershey

It might seem like a cop-out to point fingers at any one chocolate company just because it's the biggest, but it really is impossible to talk about American chocolate without heading up the Hershey highway.

An apt metaphor. Again, this is much like how Big Beer managed to convince people that the watered-down, nearly flavorless, low ABV pisswater that they sell is actually good, which it is not.

The miracle of the process is that the milk remains safe for consumption and won't spoil further, but it does have the unfortunate side effect of producing butyric acid, the substance that makes rancid butter, parmesan cheese, and human vomit all smell alike. That's the reason why American chocolate has a slightly sour, pukey taste—at least to people who aren't used to it.

I'm not sure I'd be that generous in describing the taste of Hershey's milk chocolate. Stop ragging on Parmesan, though. It's awesome.

3 Americans Hate Change

Basically, in the grand American tradition, in order for everybody to have something, we had to ruin it just a little.

It's not just chocolate and beer, either. Most things made for mass consumption aren't made so that a whole lot of people like them; they're made so that a whole lot of people find them inoffensive. Other food examples include white bread, American "cheese," and the sugary stuff that passes for peanut butter. And then people with functioning taste buds end up being left out in the cold. But we're in the minority, and they don't give a fuck about minorities; they just want to sell the most product they can, which means making them tolerable to as many people as possible.

Think about it. Every time there's a New Coke, there's a virulent backlash. The company presumably put the product through a battery of taste tests to ensure it was truly superior, but because it doesn't taste "the same," a new formula can be the death of a business. By the time new preservation methods were available, "vomit" was simply the taste of American chocolate.

And again, watered-down tasteless "beer" was the result of Americans getting used to the inferior product before and during Prohibition, and it stuck around because we don't want to change.

2 American Regulations Are More Lenient

Also the American way? Not telling anyone what to do with their business even if it sucks. European and American regulations on exactly what a product must contain and how much in order to call itself chocolate are so wildly different that they result in fundamentally different foods.

Like I said, there exists good American chocolate, mostly from small artisanal confectioners. It is, however, more expensive, because quality costs. And yet even the cheaper European chocolates are pretty good compared to the plastic crap that comes out of Pennsylvania.

While I'm ranting, "white chocolate" isn't chocolate. I have spoken.

1 Hershey Controls More Than You Think

Okay, but just because one admittedly giant company decided to make trash chocolate, why does that mean all American chocolate sucks? Believe it or not, Hershey is so protective of its formula because American chocolate companies largely seek to emulate the Hershey taste, even long after better preservation methods became available.

The last time I linked to Cracked, they were ragging on another secret formula, that of Coca-Cola. The difference there is that Coke actually tastes good.

Look, I know that tastes are different; I accept that. But when it comes to chocolate, there's real and then there's fake, and the big ones in the US are absolutely fake.

Untold numbers of European expats must have been overjoyed to find a Dairy Milk bar in a hole-in-the-wall bodega only to be violently disappointed by their first bite into it because even if you find a product supposedly made by Cadbury, a leading European chocolate brand, it won't be the Cadbury you get in Europe. That's right: Hershey's owns Cadbury in America. Hershey's is everywhere. As we say in the home of the brave, tolerate it or be a huge bummer on Halloween.

And also on Valentine's Day. Guys (mostly guys, anyway) -- if you really love your partner, you will purchase real, actual chocolate for them. And if you happen to be the recipient of a heart-shaped box of waxy, fake, puke-tasting plastic "chocolates"... well, that ought to be a giant red flag: the person giving those to you is cheap and has questionable taste, presumably apart from their choice of romantic partner, and you deserve better. Buying that crap chocolate is akin to purchasing plastic roses. Don't do it.

And while I'm at it, whose brilliant, rocket-surgery-level genius idea was it to hold the Enormous Sportsball Game the night before Valentine's Day? I mean, I don't give a crap about either one, so I find it incredibly amusing, but for the fact that I just had to lay in groceries for a week because of the double whammy coming up (and it's a good thing they're not predicting snow here, or the supermarket shelves would be a vast echoing void all weekend).

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