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In some places I read that Arabic Numeral system was originally developed in ancient India but not used till Arabs adopted it. Clarity needed………

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Intrigued, I decided to try to establish the general case and then prove it using mathematical induction. As it turns out, the general case is so incredibly trivial that constructing a proof would be absurd. (Indeed, it resolves to an algebraic tautology).

BUT – this exercise didn't satisfy me because it didn't capture the idea of “adding the numbers represented by the individual numerals composing a number”. It occurred to me that in order to do this, one would have to construct a model (c.f. Gödel) of Peano arithmetic that gives you a metalanguage to mobilize the distinction between “numeral” (or digit) and “number”. Too long and too hard for me (I'm not very good at logic nor math).

If you did make such a model, then I suspect you could also generalize THAT case to all possible base-10 positional notation systems (other than the Hindu-Arabic numeral system).

Take the plant outside and let it drop all of its leaves off that it wants too, then bring it back indoors and water once a week until it starts to grow again

As you say, "the pictures are kind of blurry", but it looks rather like
Ailanthus altissima, or Tree of Heaven. It can grow up to 15 metres
in height, and is deciduous. Its origins are in China, where it was
considered valuable for its many therapeutic properties. Today, in
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Hope this is of help, otherwise perhaps you could direct your question
to a local council/park etc. where you saw the tree.

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Tools that fit the task make it simpler. Otherwise, much of your attention is diverted to the tool itself, and not available for the actual task. This can be fun provided the task isn’t pressing.One example is multiplication in the complex roman numeral system vs. (our) simple arabic numeral system.

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Lol it was the arabs who got the transported the numeric system not the muslims,but I must admit that most muslims now(arab or not) think that they had somehow managed to create the number zero.and arabs didnt steal the numeric system from the indians they just used it and theyu even gave credit to the indians by calling them hinu numerals.Give them credit for the trus.

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The Arabic numerals are the ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, by which a sequence of digits such as "975" is read as a whole number. The Indian numerals were adopted by the Persian mathematicians in India, and passed on to the Arabs further west. The numerals were modified in shape as they were passed along; developing their modern European shapes by the time they reached North Africa. From there they were transmitted to Europe in the Middle Ages. The use of Arabic numerals spread around the world through European trade, books and colonialism. Today they are the most common symbolic representation of numbers in the world.

As you can see in this article, the Europeans and others changed them as they spread across the world

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Transfinite congratulations on the 3 Doctorows' emergent complex phenomena.

In the interpretation of the alphanumeric string “Poesy Emmeline Fibonacci Nautilus Taylor Doctorow” is am particular delighted with the “Fibonacci” — and not just as Computer Science/Math geeks everywhere are. It's because of my dialogue in a lucid dream with the eponymous Pisan.

Now, as I readily admit, this blog thread and that beautiful child, are not about me (arrogant self-centered egotistical name-dropping 2-time Con Chair fool that I am) but I want to tell you something about the eponym that few people know; some of it is true; some of it is a good story of the historico-stfnal genre.

There may still be some Math Geeks who still have not read the wonderful quotation from Fibonacci in

Post, Jonathan Vos. “Arabic Numeral.” From MathWorld–A Wolfram Web Resource,
created by Eric W. Weisstein.

In December 2006 recently I wrote in a few days the first 15,000 words of a story called “Fibonacci: Superspy” which has hovered near the novelette novel fuzzy boundary while Real Life rudely interrupted me; and had just finished a chapter featuring a naval battle between galleys against King Roger II of Sicily. I'd been at a restaurant with Dr. Thomas McDonough, a student of Carl Sagan's, and a published (“discovered by Isaac Asimov”) SF novelist. He made a note when I I told him that the pirate flag “The Jolly Roger” is named after that same King Roger. As an active member of the Skeptics' Society, he condemns claims of precognition. To our surprise, a couple of days later the L.A. Times ran a book review of another spy novel set in 12th century Italy, and featuring the same King Roger II of Sicily.

I'd started writing after, in a dream, where I was watching cute flffy rabbits jump over a fence, “like counting sheep,” I said to myself, and then noticed that there had been just one of them, than another one, then there were 2, then 3, then 5, then 8, then 13, and I said a name and, in a puff of opiated-hemp-smelling smoke, there appeared Fibonacci himself (call by name daemon) who told me that I was an idiot for thinking that I knew the first thing about him, just because I'd published gems such as:

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

“Look, whether you think you're a professional Mathematician or not, and whether you know you're dreaming or not, ” he said sarcastically, “I say again that you don't know the first thing about me. To clue you in, do you remember to whom I, whom you erroneously call Fibonacci, dedicated my first book, Liber Abaci, the one that launched the Singularity of 1202, by overnight converting the 4 major sea-going hypercompeting Italian City States, and then all of Europe, to Arabic numerals, which I had learned by studying with Arabs (well, okay with a Persian who had the best Arab texts) while living in North Africa with my father, Guilielmo Bonaccio, who wished for me to co-run the family merchant/pseudo-diplomat business. But that's just a cover story, because you don't know the first thing about me.”

“But Liber Abaci,” I stammered, “was NOT the first European book to describe Arabic numerals, because Pope Silvester II in 999…”

He cut me off with a chopping hand gesture.

“Screw Gerbert d'Aurillac,” he said, “whom you didn't know squat about either until you wrote that century in your Magic Dragon Multimedia 'Timeline' in your silly little 15,000,000 hit/year web domain. I mean Singularity.”

“What do you think happened when bookkeepers were suddenly an order of magnitude more productive? Hello, Capitalism? Impact as if of PCs finally boosting office productivity? Easier to do real Math and Science? Renaissance? You ignorant clod.”

So I woke up, jotted this all down in my bedside dreambook (which sometimes has entire complicated metered rhymed poems and sometimes gorgeous equations handed to me in dreams) and found to my shock that Fibonacci had dedicated his first book to Michael Scott. “The Wizard Michael Scott” in folklore, Dante, and Sir Walter Scott's writings, who was (bringing this ramble back to the origin) at the court of Holy Roman Emperor Frederick II — as the official Astrologer.

And I got to wondering. How did the Pisan, widely travelled, highly multilingual, grokking Judeochristian AND Islamic AND Persian culture and literature, end up at the Court of the Holy Roman Empire, and not just any Emperor, but a genuine experimental biologist? And what was he really doing, including being nominally “along for the ride” and strangely deconstructed stragetic masterpiece of a Reverse Crusade?

Bingo! He was the greatest spy of all time. And he had the perfect deep cover, with such as clever puzzle-solving applications of he Chinese remainder theorem, deep familiarity with perfect numbers and Mersenne primes, as well as formulas for arithmetic series and for square pyramidal numbers (about which I've also published), and the growth of a population of rabbits, which was actually known centuries earlier, and solving a Cubic Diophantine equation to exquisite accuracy, by methods that we still haven't figured out how he knew.

Hmmmm. So pounded the keys, and began as follows (cut & pasted from Fibonacci, Super-spy
by
Jonathan Vos Post
66-page (56 pp. story) 28,300 words [including notes]
Draft 6.0 of 3 January 2007
Copyright (c) 2006 by Emerald City Publishing
….

Call me Fibonacci.

Excuse me for a moment while I wash this blood from my
hands.

Courtiers of my patron, the Holy Roman Emperor
Frederick II, call me Leonardo Pisano. My father,
Guilielmo of the Bonacci family, rest his soul, called
me Leo. In Pisa they call me Bigollo, which means
good-for-nothing, absent-minded or a traveler. But you
and I don’t know each other well enough for you to
address me by other than Fibonacci, to properly
respect my family. A contraction of Filiorm Bonacci
(‘of the family of Bonacci’).

Did my countrymen wish to express by this epithet
“Bigollo” their disdain for myself as a man who
concerned himself with questions of no practical
value, or do they suspend judgment and simply use the
word in the Tuscan dialect, where it means a
much-traveled man, which I most assuredly am?

I arrived in Bugia just in time to witness a murder,
which led me to who I am today, by the most unusual
path.

There. It may only be the blood of a rabbit, but I’d
hate to set the court a-twitter with more slanders.

2. Bugia, 1190

Bugia is one of North Africa’s most beautiful cities.
From where young Fibonacci stood, in a hodgepodge of
Tuscan garb with African adornments, an orange in one
hand, he could see down to the sparkling azure sea,
and the thrusting cape. Turning his head, he could
see up scrubby Mount Gouraya, crowned by a cloud that
seemed caught at one corner by the rocky peak.

The city, redolent of fish, figs, frying vegetables,
dung, lemons, and spices, crossed and recrossed itself
with narrow streets never making a straight line,
always climbing, crowded with people costumed from a
hundred kingdoms.

The dirty, boat-strewn and net-littered beach
momentarily drew his mind across the Mediterranean to
Marseilles, and back to Italian fisher villages of his
younger days. Except for the fact that this place was
all Africa, not European in any way.

He turned the corner of the plum-sellers, strode past
a slave carrying a huge barrel of olives, dodged a
merchant carrying a chattering monkey, and spotted the
shop where the Persian book was being copied for him.

He slipped inside, bowed to the owner, who waved him
to seat himself on an elaborate rug atop a pile of
wool and hides.

“I bring you silver, sir. Have you finished the book,
as the messenger said?”

“Indeed, esteemed son of Guilielmo of Bonacci. I send
your father blessings on his honesty and wisdom as
Collector of Customs. His predecessor…” the owner
spat into a brass bowl, “… was known for his
bottomless greed. Your family honors your kingdom.”

The owner called in Arabic to one of his sons and
nephews, who rushed from behind a curtain to hand the
owner a splendid book, bound in leather worked with a
Persian palace scene, and handed it to Fibonacci.

The young man hardly glanced at the binding, but
flipped it open to a page of numbers and flawless
calligraphy.

“Yes! Magnificent! This is indeed the Hisab al-jabr
w’almuqabalah, just as the priceless one of my
teacher.”

Another of the owner’s family brought and poured a cup
of honey-sweet mint tea. Fibonacci took a polite sip,
a second, then set the cup down with a click onto the
inlaid tray.

“You are wise beyond your years to have a schoolmaster
who follows the prophet Mohammed, blessed be his name,
and worships Allah the Merciful. With all due respect,
your countrymen know not what they miss by following
only the lesser prophet Yeshua.”

“My teacher has opened my eyes indeed. This book
(beautiful job!) will change the world, perhaps not so
much as your Qoran. Imagine. The Persian
mathematician Abu ‘Abd Allah, Mohammed ibn Musa
al-Khowarizmi, Father of Abdullah, Mohammed, son of
Moses, native of Khwarizm, astronomer to the Caliph at
Baghdad, wrote it 375 years ago, and his al-jabr, or
Algebra as some call it now, will live forever. His
way of building the truth, one step at a time, as an
architect builds a castle, as a woman follows a recipe
for cooking a feast. How unlike anything else in the
wide world.”

The owner made a gesture over his head. “What can a
humble seller of curious goods such as myself know of
such matters? I can count sheep, count my wives,
count my children, and count a handful of coins. My
days are numbered, but Allah alone knows the number,
blessed be his name.”

Fibonacci got the hint, blushed to have forgotten the
mercantile protocol briefly, so absorbed was he in the
book. He drew the bag of silver coins from under his
blouse, with a muffled clang, and handed it to the
owner. The owner, as a gesture of trust to Guilielmo,
took it with a bow, and did not open it to count.

“And here is something from myself, not my father,”
said Fibonacci, and pulled a thin gold ring from his
finger, handing it to the owner. “What is jewelry on
the outside, compared to your hospitality, and the
treasure of understanding in the soul.”

Another sip of tea, waving aside the hookah, and
making the least smalltalk necessary to respect the
transaction, Fibonacci stood, bowed again, and stepped
from the shop back into the golden sunlight.

Suddenly, a man dressed in yellow and scarlet cloak
threw up his hands and screamed. A second man,
turbaned, his face curiously tattooed, stabbed the
first man in the belly with a curved steel dagger. The
victim screamed, gurgled, fell to the dirt. “Scotus
will know!” the dying man said. The killer stabbed
again, to the heart. The victim pissed himself,
spasmed, and lay still.

Fibonacci was squatting in terror behind amphorae of
wine and oil, hoping that he had not been seen.
Neither killer nor killed had ever looked in his
direction, but eyewitnesses had a habit of being
washed up dead upon the beach, drowned, or with their
throats cut.

“Scotus?” he wondered. “The Scots have no trade
here.”

He was awake late into the night, by firelight,
reading the luminescent mathematics, so flawlessly
copied. But when he finally slept, he dreamed not of
algorithms, but of bloody daggers.

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some of them look like tire-tread patterns. Some of the unique timekeepers cost the same as an expensive car. In general the price of the watches from the list vary from 600 dollars to 200 000.
1) Swiss chronograph Chanel J12 Superleggera heads the list. The style of its dial reminds classic Italian super cars' dash board. The watch is made of high quality materials and costs 6000 dollars.
Rolex Submariner watches
2) Cartier Roadster ranks number 2 in the list due to its sporty design. And what is interesting about the Cartier watch, its creators did not collaborate with car producers as some other companies creating automobile-inspired timepieces. The model is sold in the range of 7000 dollars. dropshipment watches
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4) TAG Heuer Carrera is the creation of a world known Swiss company that offers simple but very precise watch for 3000 dollars. Famous racers Jeff Gordon and Juan Pablo Montoya are among their owners.
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5) Pirelli Pzero is a unique watch from the tire producer. Besides its bright sporty design, the watch is offered at a very attractive price of 800 dollars.