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 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.

March 16, 2024 at 10:52am

#1066354

Substandard

Ironically, the tagline for The Takeout, which provides today's link, is: "Food is delicious."

Everything Tastes Worse Than It Used To
You've heard of shrinkflation, but what about skimpflation?

You know what else is inflating? My rage at portmanteaux. Every time I see a cutesy one, it's like I have an anger bar (like a health bar in a video game) and it just keeps getting fuller and fuller. I call it... rageflation.

It's okay when I do the portmanteau.

If you swear your go-to snacks and candies all seem to taste different—and worse—these days and you can’t quite put your finger on why, you’re not alone.

I mean... it is supposedly true that one's sense of taste dulls with age.

Business Insider has taken a deep dive into food manufacturers’ increasing adoption of what BI calls skimpflation or flavorflation, aka modifying recipes in order to (you guessed it!) maintain or increase profit margins.

Oh, good, now I know what entity to blame for the maddening portmanteau. Call it what it actually is: enshittification.

Ingredient costs, obviously, are a huge factor in the pricing of any consumer product.

So are employee wages, which is why we replace them with robots.

Business Insider cites an instance of Conagra reducing the fat content in its Wish-Bone House Italian Dressing by 10%, replacing it with additional salt and... water.

On the bright side, we're running out of fresh water, so the price of that is going to go up, too.

In 2013, Breyers, the ice cream of my childhood, had to legally change the labeling of its products from ice cream to “frozen dairy dessert.” Why? Because the company had reduced the amount of dairy fat in its product to the point that it didn’t legally qualify as ice cream anymore.

Good to know there are standards. I eat "frozen dairy dessert" so infrequently that I'd probably never notice. Part of the reason is that I have cold-sensitive teeth. But once a "quart" of ice cream became 3.75 cups or whatever, I quit buying it. Apologies for the shitty measurement system to anyone using a logical one; what you need to know is that one quart, which is a bit less than a liter, is equivalent to four standard cups.

Speaking of logical measurement systems, I remember when soda was sold in half-gallon bottles (and they were made of glass, prone to shattering, and heavy). At some point, they switched to 2-liter plastic bottles. As I noted, a liter is slightly more than a quart, and 2 liters is thus more than a half-gallon. Again, apologies to metric users: it's called a quart because it's a quarter of a gallon, okay?

I'm not really sure why. Perhaps they were thus able to increase the price beyond what it cost them to include those few extra drops in the bottle. If so, that would be one way to sneak a price increase past us: give us a bit more of the product, while charging disproportionately more.

I digress. This is about (ugh) skimpflation and not (blech) shrinkflation.

The most egregious example of so-called skimpflation we’ve seen recently was October 2022, when Conagra dropped the amount of fat in its dairy-free Smart Balance spread from 64% to 39%, which meant water became the most plentiful ingredient in the product.

I guess someone there found a smarter balance on their accounting spreadsheet. My solution? Use butter.

As an aside, I'm going to complain about English muffins (or, as I believe the English call them, muffins), which are one of my favorite foods. I even use them for hamburger buns. For a while, though, the only kind sold by my go-to grocery store was Thomas', so I didn't buy them (or I made a special trip to Whole Paycheck to get the good kind). Thomas' is to English muffins as Lender's is to bagels; that is, a piss-poor replica. Unsurprisingly, those are now both products of the same soulless corporation (aptly named Bimbo). Bread is food; everything else is a condiment.

Anyway, more recently, the nearby grocery store started selling its store-brand English muffins. One time, they were out, and the shopper (yes, I get groceries delivered, because I am remarkably lazy) subbed Thomas'. Not only are they inferior in taste, but I noticed that they seemed to be quite a lot thinner than they used to be, so thin it took hours of careful work with specialized tools to split them without destroying them. Now, that could have been my own perception, colored by comparison with the much heftier thickness of the store-brand muffins (which, I should note, are also cheaper), but it could also have been (gag) shrinkflation.

Either way, now I have to include a note with my delivery order: "DO NOT substitute Thomas'." I'd rather go without than deal with that bullshit.

In conclusion, however, the headline is wrong: Not everything tastes worse than it used to. Beer, for instance, has vastly improved in quality with the advent of craft breweries. More expensive? Sure. But worth every penny.

Well. This discussion didn't lower my rage bar. I'm going to go eat an English muffin with butter.

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