Arabic numerals

INTRODUCTION: After 800 AD the 2,800 year old ciphered numerals and finite unit fraction numeration system began to be modified in Arabic Ghobar script. The new Hindu-Arabic numerals recorded an algorithm that encoded rational numbers as 2-term and 3-term unit fraction series. Pope Sylvester in 999 CE popularized the numeration system by requiring its use in Latin schools.

By 1202 CE a fully developed Hindu-Arabic numeration, arithmetic, algebra, geometry and weights and measures system was published by Fibonacci Liber Abaci#71#>http://planetmath.org/encyclopedia/LiberAbaci.html). The Liber Abaci formally replaced the 2,800 year older ciphered Greek and Egyptian numeration system is ways that modern scholars need to read closely to appreciate the subtle changes.

The phased-out ciphered Egyptian, Greek and Hellene number system scaled n/p by LCM m to mn/mp created concise 2-term, 3-term, 4-term and 5-term unit fraction series by inspecting the best divisors of mp that summed to mn. The best unit fraction series were often not easily found.

After 800 AD the non-ciphered rational number system scaled n/p by LCM m in a subtraction context

(n/p - 1/m) = (mn -p)/mp with (mn-p) set to unity (1) as often as possible.

An algorithm found 2-term and 3-term unit fraction series in easier ways compared to the older system.

The algorithm set (mn -p)= 1 when possible to 2-term unit fraction series. At other times the algorithm demonstrated n/p recorded in a 3-term unit fraction series (example: 4/13 = 1/4 + 1/18 + 1/468).

MAIN POINTS: Arab mathematicians adopted Hindu 1-9 numerals around 800 CE. The Arab innovation replaced Greek and Hellene ciphered numerals by adding East Indian base 10 numerals in a modified base 10 unit fraction system. The Greek and Hellene ciphered numeration system followed a 1,500 year older hieratic Egyptian system that mapped numerals on a one-to-one basis to (sound) symbols. Egyptian scribes employed a formal zero, though not positional, an awkwardness that continued with Hindu-Arabic numerals after 800 AD.

The medieval 1-9 numeration system commenced in 800 AD motivated Europeans to adopt Arab numerals, Arab algorithms, and non-positional zero elements. Pope Sylvester, 999 AD, offered an math edict, to adopt Hindu-Arabic numerals and Arab and Greek mathematics. Fibonacci's Liber Abacihttp://liberabaci.blogspot.com/, 1202 AD, reported Arab numeration and simple algorithm based arithmetic system that followed the 200 year old edict.

After the closure of the Silk Road in 1454 CE, Fibonacci's 250 year old arithmetic book dropped out of use. About the same time Liber Abaci math was translated by Nicolas Chuquet in 1453. Chuquet used a lattice multiplication, double false position, Diophantine indeterminate equations, Babylonian square root, algorithms, and aspects of the Chinese Remainder Theorem methods that arrived years earlier via the Silk Road (as documented by Needham). Zero as a positional number was added 130 years later within a new numeration system and algorithm that encoded rational numbers by the binomial theorem. The well known definition of $$n^0 = 1$$ was an element. Construction details of the modern base 10 decimal system was recorded in 1585 CE by Simon Stevin. Stevin rigorously used zero in two books, one for science, and one for business, as an exponential positional place-holder. Both books were approved by the Paris Academy. Several scholars have given credit to Hindu-Arabic numerals 800 AD introduction spread to Europe and was popularized for 250 years in Fibonacci's Liber Abaci.

Elements of Napier's Bones, and Arab- Hellene lattice multiplication method were noted by Fibonacci. Napier popularized the new base 10 decimal system by adding logarithms. Napier's numeration publications facilitated several science activities, including Galileo's 1609 astronomical work and logarithms were used to facilitate Huygens' telescope.

The last surviving Ghobar unit fraction document was written in 1637. The text pointed the way to Mecca from Morocco. Modern Arabic script replaced Ghobar script in the 17th century, a set of actions that formally ended 3,700 of continuous unit fraction arithmetic and mathematics use in the Egyptian, Greek, Arab, and medieval math worlds.

BACKGROUND: Egyptian and Greek scribes ciphered numbers onto sound symbols (letters) on a one-to-one basis. Zero was used but was not a positional idea. Babylonians also used a practical zero (Neugebauer) near 2050 BCE when Egyptians used the word sfr for zero in double entry accounting and other applications.

Greeks mapped numerals and rational numbers onto Ionian and Doric letter symbols by adding ('), ie. 1/2 = to beta', 1/3 = gamma', and so forth, recorded as unit fraction series. The Greek zero symbol recorded an oval topped with two dots, a set of notations that began to be replaced with the arrival of Arabic 1-9 numerals.

CONCLUDING COMMENTS: After 800 AD a subtraction context ciphered rational numbers (n/p - 1/m) to (mn -p)/mp to (mn -p) recorded 2-term and 3-term series. Fibonacci reported three Arab unit fraction notations. The most popular Liber Abaci notation was translated by Nicolas Chuquet in 1453. Chuquet used a lattice multiplication, double false position, Diophantine indeterminate equations, Babylonian square root, algorithms, and aspects of the Chinese Remainder Theorem methods arrived 1,500 years earlier via the Silk Road.

In 1585 AD medieval unit fraction arithmetic was written in Hindu-Arabic 1-9 numerals ended in Europe. The medieval Ghobar script version of the unit fraction system contained arithmetic words and math operations also ended when modern Arabic script replaced Ghobar script in the 17th century.

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