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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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October 11, 2021 at 12:08am

#1019101

Bubble Rap

Well, yesterday's entry was about hamburgers (mini-contest results below). So it's only fitting - though random - that today's is about beer.

The bubbles in your pint of beer are a scientific marvel
Pouring a chilled beer into a tilted glass produces about 1.5 million bubbles

This gives me a chance to trot out one of my favorite words again: zythology. While the article focuses on effervescence, that's part of what makes beer great.

Beer is one of the oldest drinks that humans have made.

Praise Ninkasi!

And in a recent study, a pair of scientists from the Université de Reims Champagne-Ardenne...

They weren't kicked out for studying la bière instead of le champagne? (To be fair, probably to avoid losing whatever passes for tenure in les universités françaises, the article goes on to note that they also studied this in champagne. I also love wine, sparkling or otherwise.)

...have drawn attention to a dazzling event that occurs every time we pour a cold one in a glass, by focusing on a key ingredient, carbon dioxide (CO₂).

I was going to quibble over the use of the word "ingredient" there. When magic yeast eat sugar, they piss ethanol and fart carbon dioxide. So it's not an ingredient in the sense of "additive," not like the classic water-malt-hops-yeast; it's more of a byproduct. Well, like I said, I was going to quibble until I remembered that a lot of breweries enhance their products via carbonation, much as soft drink manufacturers do, so I guess it really is an ingredient and I'm glad I thought about it before I got pedantic.

By the way, one of the things that stuck with my from my background in chemistry is a weird thing about dissolved gas in a liquid. Most people are aware that heating a liquid usually increases the solubility of solid ingredients. For example, if you're making simple syrup for cocktails, you need to heat the water so the sugar will fully dissolve. When it cools down again, it usually stays in solution; this is called "supersaturation." Well, it's the opposite for most gases: the colder the water, the more dissolved gas it can hold. This is one reason why you want to open warm beers very, very carefully (if at all; warm beer sucks, though some are best at cellar temperature).

Beer is a liquid supersaturated with carbon dioxide gas. While contained in a sealed bottle, a high pressure keeps CO₂ dissolved in the liquid in a larger amount than would normally exist at atmospheric pressure. But when the bottle is opened, CO₂ quickly escapes into the air producing a wave of bubbles and foam.

This is also why you don't shake carbonated beverages. One TV/movie trope I absolutely despise is the one where they're celebrating, so they shake the fuck out of a bottle of champagne before opening it, causing a fire hydrant spray of foam. This is an atrocity, a blasphemous practice that is disrespectful to the manufacturer and, worse, to the beverage itself.

The scientists compared bubbling in beer with the same feature of champagne. They found that pouring a chilled Heineken over a tilted glass produced about 1.5 million bubbles – not very different from champagne, which produced slightly over 1 million bubbles.

Right now you may be wondering, "why does sparkling wine, including champagne, have bubbles while regular wine doesn't?" I dunno, and I can't be arsed to look it up. I imagine it has something to do with the strain of yeast involved.

However, the number of bubbles produced was dependent on the size of microcrevices in the glass. Microcrevices are sites on the surface of a glass wall that are only visible with a microscope, and they are where bubbles originate. They can span from 1 to 10 micrometers in size, roughly as small as a grain of sand.

This is misleading. The 1 to 10 micrometer range might overlap with the size of sand grains (whose definition is, as I noted in a previous entry, entirely arbitrary), but not according to the sources I found. Look at it this way: We can see down to about 0.1 mm

. Most of us can distinguish individual grains of sand, so they're generally bigger than 0.1mm. One micrometer is 0.001 mm - two orders of magnitude smaller. So something's off here. (We also usually can't see microcrevices because, duh, they're "micro.")

Another key finding was that champagne contained about five times more dissolved CO₂ than beer. And surprisingly, the amount of CO₂ dissolved in beer was only a little over that of carbonated water. In addition, beer bubbles were smaller than champagne bubbles and move through the drink differently. Smaller bubbles are preferred in champagne, according to Liger-Belair, “Mainly because small ascending bubbles rise more slowly which is more elegant.”

Oooh, elegant. Screw elegance; I just want a fine-tasting fermented beverage.

But yeah, effervescence is one factor that goes into my enjoyment of a beer. Different beers have different bubble profiles. There's a Belgian beer I know that forms a thick, foamy head that lingers. Other beers lose foam quickly, but keep the effervescence. Some beers don't ship well, and the bar will infuse it with nitrogen right there at the tap; this is usually done for darker beers, and the resulting cascade of minute foamy bubbles is an integral part of the experience. You can watch them rise to the top in waves. This is especially fascinating after you've already had a few. It does alter the taste, though, making a beer more creamy. I like it, but some people prefer the larger bubbles from CO₂.

Mini-Contest Results!

You know, sometimes I don't think my cunning plans all the way through. I have a routine medical procedure scheduled for this week, one that requires me to follow a certain diet for a few days. You know what I can't have on this diet? Raw vegetables. And a lot of cooked ones, like onions. Which makes most of the mouth-watering burger recipes the Forbidden Fruit, at least until Thursday evening. So why did I ask for hamburger recipes? Like I said, sometimes I don't think my cunning plans all the way through. Damn, I'm craving a burger. Sure, I could eat just cooked ground beef on a roll, even with cheese, but that's not going to cut it right now.

I loved all the various burger descriptions and recipes. Some of them are more appealing to me, personally, than others are, but I could tell that you all loved what you were writing about, and that's cool. You all made it tough to pick just one, but for today I'm going to go with ForeverDreamer

for his awesomely described ideal burger, even though I'm not a fan of ramps (I did say I wasn't going to necessarily pick one I'd personally like). But all y'all gave me some good ideas and different perspectives on the culinary wonder that is the Hamburger.

And made me very, very hungry.

I like mine with lettuce and tomato
Heinz 57 and french fried potatoes
Big kosher pickle and a cold draft beer
Well, good God almighty which way do I steer

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