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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QPdoll

about 11 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

about 14 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

2 Like

October 9, 2023 at 9:11am

#1057044

I Am The Law

I didn't fact-check any of these. It's just something fun from Cracked.

5 Laws Designed to Screw Over One Guy in Particular

One person does something bad, and so we pass a new law, to forbid anyone else from trying that same thing. That may not be the most thoughtful path toward creating a legal code, but it’s a routine way of doing things.

Which is how you end up with laws like "No baton twirling within 20 feet of park benches." Introduced at the last city council meeting by an elected official with a bandaged head.

Sometimes, though, when we pass a reactive law, we’re not worried about that crime becoming a trend. We just want to keep our thumb on that one guy.

As far as claims to fame go, you could do worse than "inspiration for a law."

5. Ohio’s Urine Collection Law

Part of their pee-nal code.

In 2008, a 56-year-old man named Alan David Patton was caught collecting urine from a bathroom in a park in Dublin, Ohio.

Okay, ew.

Authorities charged him with criminal mischief. This broad and ill-defined law landed him a maximum of 60 days in jail, which really didn’t seem enough for this weird freak. So, they came up with a new law. Ohio now made it illegal for anyone for to collect “bodily substances without consent and for non-medical purposes.” The new penalty for this new crime? Six months in prison.

Urine trouble now.

4. The Right to Detain Greg

Australian man Greg Kable stabbed his wife to death in 1989. Not cool, Greg.

In case it's not clear, wife-stabbing was already illegal in Oz.

The sitting Parliament passed a new order called the Community Protection Act. If a prisoner due for release was deemed a danger to the public, said the order, they would keep him in prison until they were satisfied he wasn’t... Fortunately for all future prisoners, New South Wales amended the act to limit its scope. Now, it would only apply to Greg Kable, specifically.

"Whataya in for, mate?" "Being Greg Kable."

3. Banning One Kid from Talking About Grass

This one's way more interesting than just that bit.

Sometimes, this made sense. For example, a 26-year-old was brought up for repeatedly playing “Do They Know It’s Christmas” to annoy his neighbors, an offense that ought lead to life in prison.

Nah. 20 years, max. Life sentences should be reserved for "Simply Having A Wonderful Christmastime."

An ASBO from 2003 forbade an 87-year-old man from being sarcastic to his neighbors.

I'd be boned.

2. An Anti-Armstrong Law

The burgh of Langholm in Scotland has a law on the books that says any Armstrong who enters the town must be hanged.

Maybe we should arrange a bicycle race through there.

The Armstrong was Johnnie Armstrong, who lived in the 16th century. The English remember him as a criminal due to his habit of raiding England, while the Scottish remember him as a folk hero, due to his habit of raiding England.

That quote by itself was why I had to blog about this article. Bonus, though:

We’d probably never know about this old law. Except, in 1972, Langholm received a famous visitor: astronaut Neil Armstrong.

"We must obey the law. Quickly, now, hang... a portrait of Neil Armstrong in the museum!"

1. The Man Who Worked His Butt Off

This one's a stretch (pun absolutely intended). It's not really a "law."

As of 2016, Britain’s National Health Service had a certification for something called a “rectal teaching assistant.” This was someone employed by the government to visit medical schools, so doctors in training could learn how to conduct prostate exams by practicing on his well-documented body.

And I have no doubt he was the butt of many jokes. Rectum? Damn near killed 'im.

And with that, I'm done. If I were more ambitious, I'd double-check all these, but more importantly, I'd try to find examples from outside the English-speaking hegemony. I'd bet Germany has some fun ones, for example.

7 Like • Comment • View 2 comments

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