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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Write_Mikey_Write!

a day 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

a day 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.

October 10, 2019 at 12:18am

#967567

Gindex

https://www.nbcnews.com/business/consumer/pour-decisions-better-educated-you-are...

Pour decisions? The better educated you are, the more you drink
Americans with a graduate degree spend 10 times more per year on booze than high school dropouts do.

Let's get the obvious out of the way: People with a graduate degree, on average, have more money to spend on stuff than high school dropouts (again on average). This is addressed deep within the article. Well, if you can call short clickbait "deep."

What I did not see addressed and seems obvious to me: Booze prices vary wildly. We know about "Two-Buck Chuck," the discount wine. I wonder what the British equivalent is? Two-Quid Sid? Anyway, I've also seen a bottle of wine priced at $23,000.

An aside: a common trick in restaurants and such is to have an item with a decoy price. It's a psychological thing that works like this. Say you have a menu with two bottles of wine; one is priced at $10 and the other at $30. People tend to think the $30 bottle is way too expensive and buy the $10 one (which probably cost the restaurant $5). But if you add a $60 bottle of wine to the menu, suddenly the $30 seems not as bad. A bit decadent, maybe, but at least it's not $60, right?

So that $23K bottle of wine is the most egregious use of the decoy price I've ever seen. Some of the other bottles were priced in the $100-$1000 range - way, way more than I would ever spend on a bottle of wine, because I figured out a long time ago that once you get wine above around $50, I can't tell the difference, and I'm not one to buy expensive stuff just to show off. That said, I've bought scotch in that price range - because with scotch, I can tell the difference between a $50 bottle and a $200 bottle.

That was in Vegas, by the way - lots of things are overpriced in casino restaurants because of comps and to suck the money back out of the occasional exuberant winner.

But I'm getting off-track. My real point here is that people with more money often choose to buy more expensive booze. Maybe not the $23K bottle of wine, but maybe a wealthier person would decide to go for quality over quantity, as I do. I'd rather spend $100 on a bottle of scotch and savor it slowly over the course of months, than blow $5 (or whatever) on a bottle of rotgut every day, because I'm not using booze to commit suicide.

In other words, there is no support for the conclusion in the headline: "the better educated you are, the more you drink." The only correlation seems to be with the amount spent on alcoholic beverages, which, as I've noted, doesn't correlate to the volume of beverage imbibed.

Or take beer, for example (better yet, give me some). I just looked up Buttwiper at Wal-Mart, and it's $16 for a 24-pack of 12-ounce cans. Doing some quick math, that's 67 cents a can or about 5.5 cents per ounce. Dirt cheap. Meanwhile, today I ordered a premium craft brew (albeit of much higher ABV) at $9 for a 10 ounce pour. That's 90 cents an ounce, obviously, which is over sixteen times the cost of Bud at Wal-Mart. Even if you compare ABV, Bud is at 5% while the delicious stout I ordered is around 10%, so if you're going with alcohol content, that's still an eightfold difference.

You can do the same thing with wine, whisk(e)y, gin... whatever. There is a weak correlation between price and quality of any kind of booze - you get some outliers, sure, where cheaper stuff can be of good quality, but on average, cheap is rotgut and expensive is nectar of the gods.

Consequently, if you're only going to go by amount spent, it says nothing about the quantity of alcohol ingested by richer, educated people compared to poorer, less educated people. Mind you, I'm not ragging on the poor or throwing shade on the rich here, but it seems to me that a poor person who wants to drink can find cheaper options, so it could be that the more economically challenged drunk can spend less to get the same level of inebriation - if that's what they're going for.

So - yet another example of not being able to take a headline at face value.

While money is just one factor, those averages could also be affected by people who don’t drink. A Gallup survey found that those who do not pursue higher education are less likely to drink. Eight in 10 college graduates said they sometimes drink, while just half of people who reported a high school diploma as their highest level of education said they drink.

Now that is an interesting finding. You have this stereotype of the working-class guy sitting on his couch slugging Buds, but it's apparently more likely that the college professor is sipping scotch whilst grading papers or whatever it is professors do.

However, beer and wine companies may have a problem. Overall, worldwide alcohol consumption fell 1.6 percent this year, according to IWSR, a market intelligence group focused on the global alcohol industry.

Yes, beer and wine companies may have a problem, but it's not a paltry 1.6 percent decline - which is offset by the approximately 1.1% annual increase in world population, which the article leaves out. No, the problem is competition - the market is already saturated, in both senses of the word.

Alcohol companies are now seizing an opportunity to appeal to people who are focused on health and wellness by offering lower or non-alcoholic versions of their adult beverages. Heineken sells a brew called 0.0, which tastes like beer, but without any of the alcohol.

You do you, but... no, thanks. I'll just drink water.

While beer and wine were down, IWSR reported growth in spirits. The biggest driver was gin, which posted an 8.3 percent increase over the past year.

I propose a new index of world happiness: the Gindex. There are premium gins, but if you want to get drunk fast and cheap, there's plenty of crappy gin out there that can get you drunk. The marvelous thing about alcohol is that you can use it to celebrate, or you can use it to mourn (the latter is not recommended, even by me, because I embrace an alcohol-positive lifestyle, but it happens). So I think - and this is unsupported by any evidence, but I'm going to go with it for now - that the more gin that gets imbibed, the worse off people think they are. High gindex would be correlated with lower quality of life.

Still, I do like my gin, so I'm happy to see it staging a comeback.

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