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 15 hours 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 ✍️

about 18 hours 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

about 21 hours ago
In response to "Gray Matters"

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

2 Like

tj-Merry Mischief Maker

a day ago
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

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

May 4, 2024 at 8:21am

#1070504

Overdraft Dodgers

I have way, way more links from other sources in my queue, but hit another Cracked article today anyway. Either it's just one of those things with random numbers, or the Universe is trying to tell me something this week. I'm going with the former. At least this one's not a countdown.

This Is the Monster Who Invented Overdraft Fees

While there were obviously worse monsters in history—certain dictators come to mind, as does whoever used the next-to-last bit of toilet paper and didn't replace the roll—this guy's nearly up there with them. Of course, they tease us by not revealing the name until we scroll far enough down, and whoever invented that bit of web optimization also needs to be called out and shamed.

Spoiler: it's a guy named, appropriately, Hogg.

The overdraft fee. A trap disguised as convenience, so nefarious that it feels like it was invented by Mephistopheles himself.

Maphi... Meppi... Mephi... the devil had better things to invent, like blister packs and those hard plastic shells that take power tools to open.

Not especially surprising, given that currency, something originally invented to make commerce easier, is now basically a way to use math to break a human’s spirit over the course of their lifetime.

To hear some people talk about it, the only math that doesn't cause them great anxiety is when the store offers a 30% discount on some gadget in a hard plastic shell. Why hasn't anyone made bulletproof armor out of that stuff yet?

Any discussion or criticism of overdraft fees always results in the same defenders bubbling up from the cracks like ichor: People who are frustrated that bank accounts are private information, because they want everyone to know how financially responsible and smart they are. “Maybe you should keep track of how much money is in your bank account, and you wouldn’t overdraft,” they offer smugly, like a straight-A student tsk-tsking at the kid who forgot his homework.

Bank accounts, like health problems, are only private information if you want yours to be. For instance, I have a checking account with exactly $500.00 in it. I've never overdrafted it in the 20 years or so that it's been in existence. There, I've shared private information.

That said, that "advice" is simplistic and annoying, and ignores Rule #1 of finance: Banks will find a way to screw you.

Let’s also make it clear here, in case you were kicked by a horse and are still defending them as a valuable convenience: Banks offer overdraft fees because they make a shit ton of money off of them.

The overdraft fee can also be considered interest on a loan. Sometimes, that interest is so high the mob looks at it and goes, "Geez, why can't we charge that much?"

2019, for instance, they made $11.68 billion from overdraft fees alone. Which is the kind of cash that inspired some banks to reorder transactions, deducting big purchases first, so that they might have a chance to hit multiple overdraft fees from some sap by rattling off five unaffordable coffees once they were at zero.

They also like to process debits (checks, card purchases, transfers out) before deposits. See Rule #1 of finance, above.

While overdraft fees seem like a modern cash grab...

Their origin goes all the way back to 1728, and a Scottish merchant named William Hogg. He, together with the Royal Bank of Scotland, worked out the first ever overdraft fee, so that he could conduct business more effectively if he was waiting on a tardy payment.

While the link on his name in the article is broken, I'm here for you: {xlink:https://freepages.rootsweb.com/~hogg/genealogy/famoushoggs.html}this page} lists several famous Hoggs, including Ima, whose parents would be roasting on spits in Hell, if Hell actually existed, for naming her that; William Hogg is on there, with a link that confirms that, indeed, he's the guy who helped invent overdraft fees.

But Hogg's influence is minor, analogous to whoever invented fire when compared to whoever invented napalm.

The problem being, of course, that reliable customers don’t bounce enough checks to make it a revenue stream. Here is when the crown of hell is lowered onto a man’s head: William H. Strunk, who in 1994 came up with the modern overdraft system, “available” to every customer, and consisting of a per-transaction charge.

What is it with guys named William, anyway? Are they just mad because no one likes to receive Bills?

The final piece of this devil’s triad was the widespread use of debit cards. With people using debit cards for every manner of purchase, instead of larger bank debits and ATM withdrawals, suddenly banks were presented with a smorgasbord of tiny purchases to swat back with a new, double-digit fee attached.

Even there, one might think, well, if there's only $17 in your account, and you try to make a $20 purchase with your debit card, shouldn't that just be declined? Well, maybe. But that assumes the bank doesn't want to collect an OD fee, which it definitely does, in accordance with Rule #1. Also, maybe you've just checked your balance using your handy mobile device, and it says you have $117, but what it doesn't say is that the $100 check you wrote to the babysitter hasn't cleared yet.

There's certainly an aspect of personal responsibility involved, but no one's perfect, and imperfection shouldn't come with a 2000%+ interest rate.

Luckily for us, their day in the sun seems to be ending, as the government is taking on overdraft fees directly.

The government? The government that's run by bankers? That government?

Obviously, all this applies in the US. In other countries, your kilometerage may vary.

4 Like • Comment • View 1 comment

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