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:

tj-Merry Mischief Maker

about 7 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

about 12 hours 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

about 14 hours 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 ✍️

a day ago
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

a day ago
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

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

April 24, 2024 at 1:02am

#1069547

Tipped Out

Yes, the article I'm sharing today was one of my inspirations for the Comedy newsletter editorial that just came out: "Keep the Tip"

. It's coincidence that my random number generator picked it out of my queue today.

What is the new etiquette for tipping?

I've also done other blog entries on the subject before, but this is a relatively new take, from NPR.

Businesses that never seemed to ask for a tip before — like grocery stores, self-checkout machines and fast food restaurants — are now asking for one these days.

As I noted in the editorial, I can't personally confirm the self-checkout tip chutzpah, but this is not the only source that asserts that it's happening. It's rare to find begging that blatant and uncalled-for.

If a business you don't expect to ask for a tip is suddenly asking you for a tip, what should you do?

Ideally, take your business elsewhere.

But Shubhranshu Singh, a marketing professor at Johns Hopkins University, likes to leave a 10% tip. If an establishment is asking for a tip, it's often an indication that the workers there are not getting paid a minimum wage.

If the workers aren't getting paid a minimum wage (outside exempt employees such as restaurant servers, who are some of the few that should be tipped), that business needs to face consequences for it. How that would work, though, I don't really know.

Some businesses load their payment systems with default minimum tip options of more than 20%. If you don't want to give that much, don't worry about holding up the line to take an extra moment to select the "custom tip" option, says Singh.

In those situations, I select the "no tip" option, and definitely try to find somewhere else to shop in the future. I know it's usually not the workers' responsibility to set those defaults. Still, it's bloody outrageous.

Don't forget to tip people who you might not have a direct interaction with, like hotel housekeepers, says Singh.

I'm torn on the whole tipping hotel housekeepers thing, honestly. I've done it. I've skipped out on doing it. I feel like it should be done for extended stays, on the theory that a tipped housekeeper might go the extra mile for you and hopefully not put itching powder in your undies. This makes it more of a bribe than a tip.

Tipping is also a way to pay workers more without actually raising their wages. It allows restaurants to get more money to workers while still keeping their prices low, says Sean Jung, a professor at Boston University's School of Hospitality Administration.

But if tips are expected and customary, as with restaurants, the "low" prices (they've still gone up along with everything else recently) are a lie. Nevertheless, some restaurants have experimented with no-tip options. Usually, this means raising their prices by 20-25%. Then, when comparison shopping, a potential customer sees higher prices and balks, ignoring which places take tips and which don't. Every place I know that tried this route went out of business.

In other words, it needs to be across the board, which means either a) collusion, which, as I understand things, is illegal; or b) regulations, which are already overdone.

Tipping culture in the US is so ingrained that there's not much will to do either. And even if they did, people will still try to tip, because it's what they're used to doing.

Oh, well... at least this article, unlike some others on the topic that I've shared here, doesn't rehash the false TIPS backronym, which, as I've noted repeatedly, has been thoroughly debunked. And yet, the article is light on practical information and encourages way too much work for the consumer, who just wants a damn hamburger.

I'm not saying people shouldn't get decent pay. Quite the contrary. I'm just saying it shouldn't be laid directly on our shoulders. I'd prefer to see the actual price of things up front (and this also goes for hidden fees from businesses like airlines, ticket sellers, and hotels). Until that utopia comes into existence, though, I continue to tip waitstaff and bartenders.

2 Like • Comment • View 2 comments

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