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 13 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 16 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.

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Annette in France

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

Bienvenue en France.

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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

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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

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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!))

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September 16, 2023 at 8:01am

#1055814

Penny For Your Thoughts

Ever been told to stay in your lane?

The Windy History of Penny Lane: The Beatles, the Slave Trade and a Now-Resolved Controversy
Was Penny Lane named after a notorious slave trader? Recent protests reignited the debate

If you've been following along, you know I rarely stay in my lane. One excursion I often make is into the subject of etymology, and, in the past, I've been especially fascinated by the etymology of currency. For instance, did you know what dollars and Neanderthals have in common? Both were named after valleys in what were then German-speaking regions: Joachimsthal and Neanderthal (those are old spellings; modern orthography omits the h from the th). Joachimsthal was a silver-mining location, and the silver coins made from its product were Joachimsthalers, and it's easy to see how you get from Joachimsthalers to thalers to dollars. (Neandertal has a linguistic connection to "new man," as it's partly a Romanized version of the surname Neumann, or "new man," but that seems to be one of those apt but unrelated coincidences that happen occasionally.)

The origin of the pound (as a unit of currency) is similarly well-documented. But for something so common and widespread as the penny, well, they're not entirely sure. It's somehow related to the Roman denarius, which is why, for example, 12d nails are pronounced "twelve-penny nails." What I'm fairly certain of is that it's completely unrelated to the similarly-spelled penis, which in Latin was a word for penis. Penis.

Anyway, it gets even murkier when you have people with the surname Penny or Penney.

Apparently, one of them was a slave trader, and of course that's one of the occupations we don't name things after anymore.

An old theory linking the street to a notorious slave trader had resurfaced due to the protests surrounding the police killing of George Floyd — and a cadre of local historians discovered that their research was now thrust into the public eye.

That idea doesn't deserve the label "theory." At best, it's a guess.

“It’s been an academic debate, really. So it’s a bit of a surprise to us all, to be honest; we’re sort of taken aback. We’re not used to this larger media interest in the names of streets going back to this, you know, 17- and 1800s — it’s not the usual thing that makes the news.”

My hot take on this is: even if it was named after James Penny (for which, as the article indicates, there is no evidence whatsoever), that was so long ago, and the word is so common in other contexts, that it just doesn't matter anymore.

Besides, any metadata on that particular street changed drastically nearly 60 years ago, when it became forever and indelibly associated with the Beatles.

Following the graffiting of the signs, though, Liverpool’s Metro Mayor Steve Rotheram made international news after proclaiming the famed street name may be changed if there was evidence it was named after 1700s slave trader James Penny.

On the one hand, it's their city and they can do what they want with it. We fought and won a war so that England couldn't tell us what to do anymore; it's only fair that we don't interfere with them, either.

But, as you also know if you've been following along, I really hate to see falsehoods become accepted as truth.

Enter MacDonald and other historians, who have been researching the area for more than 10 years and claim there is no connection between Penny Lane and the slave trade.

It's remarkably hard to prove a negative. The only thing a historian can do in such a case as this, if they can't definitively show that Penny Lane was named for someone (or something) else, is to indicate that there's no known connection. So, if not some arsehole trader of human beings, what was it named after?

They, um, don't know.

According to the historian, the earliest mention of the lane was from the 1840s, when it was listed as Pennies Lane. In maps going back to the 1700s, it was merely an unnamed country road. Meanwhile, James Penny died in 1799 — plus, he already had a street named after him: Arrad Street, named for his birthplace in Ulverston, Cumbria.

Okay, but that last bit is hardly definitive. Plenty of slaveholders had multiple streets named after them here in the US.

I think the important bit is "Pennies Lane," though. Even with the very fluid spellings of the 18th and early 19th century, you'd think that it would have shown up as Penny's Lane or at least Pennys Lane, if it was connected to anyone with that surname.

“Penny Lane about that time would have been a fairly rural country lane,” MacDonald says. “So that struck me. It would be very off that a lane in the middle of the country would be named after somebody in the same way that prestigious streets in the town center would.”

You know what would make sense for a rural country lane in 19th century England? If you had to pay a toll to traverse it. Tuppence, perhaps, or thruppence with inflation. Hence, "pennies lane." But did these boffins investigate that blindingly obvious (to me) possibility?

Well, maybe. Maybe not. The article doesn't say. So don't go quoting me on the last paragraph; it's just as much a guess, albeit a harmless one, as the spurious James Penny connection.

In any event, as the article eventually winds around to telling us, it seems that "no evidence of a connection" is sufficient to keep the iconic street name around... for now. These controversies, however, have a way of coming out of remission later. If that's because some evidence is actually discovered, well, fine. But if it's just another urban legend, well, I trust the Liverpudlians will treat that with all the attention it deserves: about two pennies' worth.

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