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:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 22 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

March 8, 2024 at 8:25am

#1065851

Law and Ordering Alcohol

Booze laws can be convoluted. You think they exist to promote public health and safety, and maybe some of them do, but mostly, the laws are about who does and does not get to profit from alcohol. (In order, that would be "governments and rich people," and "you.") Leave it to Cracked to find some of the weirdest ones.

5 Perplexing Laws Over Who Gets to Buy Booze

5. In New York, Bartenders Are Forbidden from Not Giving Alcohol to Pregnant Customers

Of all these examples, this one's actually the least weird.

In general, bartenders are allowed to refuse service to customers. In the case of an intoxicated customer, they’re legally bound to refuse service. But they are legally banned from refusing service to a customer because they're pregnant, as this may be “pretext for discrimination or as a way to reinforce traditional gender norms or stereotypes.”

For starters, the header is misleading. Bartenders aren't known for "giving" alcohol. They're known for selling it, and for patiently listening to sob stories in hopes of getting a tip.

Further, while late-stage pregnancy is usually quite visible, from my understanding, some of the worst and most (eventually) visible effects of drinking while pregnant occur during the first trimester, when it's not usually obvious.

And finally, we either need to assert that all adult humans are responsible for their own decisions, or go the way of places like Texas who exert government control over the pregnant ones.

4. The Very First Campaign Finance Law Was About Banning Alcohol Sales

*in the US

In 1811, Maryland banned liquor sales on Election Day, but the goal wasn’t just keeping the electorate sober in general. The goal was to keep politicians from buying voters alcohol — as bribes. As a result, this is remembered as the first campaign finance law.

Well, this one's not that strange, either. If I'm having a hard time choosing between two evil politicians, I would definitely vote for the one who bought me a beer.

3. Happy Hour Bars That Legally Must Give Discounts on Soda

During happy hour, bars slash prices on drinks, maybe selling you two for the price of one.

The laws I'm most familiar with are, obviously, those of my own state. I still don't fully understand them, but to the best of my knowledge, in Virginia, you can't do two-for-one promotions or ever give out free booze. We do get to enjoy discounts, sometimes, and a half-price beer is functionally the same thing without forcing you to buy two drinks.

This drives up business during times that are otherwise dead, and if they’re able to make a profit after halving their prices, that lets you know just how high the markup is on drinks the rest of the time.

Markups can be high, but this is not necessarily the case. Sometimes it's about ensuring the bartender has the opportunity to get tips so they're less likely to run off and work somewhere else.

France has it’s own anti-happiness law. They do have happy hour there (in France, they call it apéro hour), but since 2019, if a bar offers discounts on alcoholic drinks, they also have to offer discounts on soft drinks.

Sadly, the law we need is to fine people for misusing apostrophes.

2. A Special Car for Drunk Drivers

Also in France, if you get caught driving under the influence, they might suspend your license, leaving you unable to operate your car.

I mean, technically, no, not unless they also impound the vehicle and put you on the "no-buy-vehicles" list. But legally, sure.

If a vehicle moves at a top speed of 28 miles per hour or less, and has an engine with an output of no more than 5.4 horsepower, it’s a quadricycle, and in France, it’s known as a voiture sans permis, a VSP. That means you can drive it even if you have no valid license, so long as you’re 14 or older and have a few hours of recorded driving experience.

On the one hand, a top speed of 28 mph (or, as I'm sure they express it over there, 45 kmh) is going to reduce, though not eliminate, the hazards of drunk driving. On the other, you're pretty much limited to city driving, and cities there tend to have bars within walking distance, so what's the point?

1. The U.S. Has No Minimum Drinking Age, Actually

Yeah, technically correct. But effectively not.

Today, all 50 states set a 21 minimum. Even so, nearly every state offers some exceptions. In states like Wisconsin, for example, people of any age can drink so long as they’re with a parent or spouse who’s over 21.

Wisconsin's drinking culture is a truly awesome thing to behold.

In Puerto Rico (which is part of America, despite what that one song from West Side Story left entire confused generations thinking), the minimum drinking age is simply 18. Puerto Rico has highways, but they gladly reject 10 percent of the funds they could get. Some things are more important.

Me? I say lower the drinking age and increase the driving age. That way you get most of the stupid out of your system before you even get behind the wheel of a car.

Better yet, hurry up with the self-driving car thing. As you know, I'm alcohol-positive. But drunk driving? Let's just say I agree more with the French on that one.

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