Roman Numeral Calculations

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Roman Numerals - Roman numerals are a numeral system of ancient Rome based on letters of the alphabet, which are combined to signify the sum of their values. The Roman numeral system is decimal but not directly positional and does not include a zero. It is a cousin of the Etruscan numerals, and the letters derive from earlier non-alphabetical symbols; over time the Romans came to identify the symbols with letters of their Latin alphabet. The system was modified slightly during the Middle Ages to produce the system used today.

Roman numerals are commonly used in numbered lists (such as the outline format of an article), clock faces, pages preceding the main body of a book, chord triads in music analysis, the numbering of movie publication dates, months of the year, successive political leaders or children with identical names, and the numbering of annual events.

Although the Roman numerals are now written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used similar symbols, of which only I and X happened to be letters in their alphabet. One folk etymology has it that the V represented a hand, and that the X was made by placing two Vs on top of each other, one inverted. However, the Etrusco-Roman numerals actually appear to derive from notches on tally sticks, which continued to be used by Italian and Dalmatian shepherds into the 19th century.

Thus I descends not from the letter I but from a notch scored across the stick. Every fifth notch was double cut , and every tenth was cross cut (X), much like European tally marks today. This produced a positional system: Eight on a counting stick was eight tallies, or the eighth of a longer series of tallies; either way, it could be abbreviated VIII, as the existence of a Λ implies four prior notches. By extension, eighteen was the eighth tally after the first ten, which could be abbreviated X, and so was XΛIII. Likewise, number four on the stick was the I-notch that could be felt just before the cut of the V, so it could be written as either IIII or IV. Thus the system was neither additive nor subtractive in its conception, but ordinal. When the tallies were transferred to writing, the marks were easily identified with the existing Roman letters I, V, X

The tenth V or X along the stick received an extra stroke. Thus 50 was written variously as N, but perhaps most often as a chicken-track shape like a superimposed V and I. This had flattened to an inverted T by the time of Augustus, and soon thereafter became identified with the graphically similar letter L. Likewise, 100 was variously H, or as any of the symbols for 50 above plus an extra stroke. The form superimposed X and I came to predominate. It was written variously, was then abbreviated to C, with C variant finally winning out because, as a letter, it stood for centum, Latin for "hundred".

The hundredth V or X was marked with a box or circle. Thus 500 was under the graphic influence of the letter D. It was later identified as the letter D, perhaps as an abbreviation of demi-mille "half-thousand"; this at least was the folk etymology given to it later on. Meanwhile, 1000 was a circled or boxed X and by Augustinian times was partially identified with the Greek letter phi. In different traditions it then evolved along several different routes. Some variants were historical dead ends, although folk etymology later identified D for 500 as graphically half of Φ for 1000 because of the CD variant. A third line survives to this day in two variants. One, (I), led to the convention of using parentheses to indicate multiplication by a thousand: the original (I) 1000, then (III) for 3000, (V) 5000, (IX) 9000, (X) 10 000, (L) 50 000, (C) 100 000, (D) 500 000, (M) 1000 000, etc. This was later extended to double parentheses.

In general, the number zero did not have its own Roman numeral, but a primitive form (nulla) was known by medieval computists (responsible for calculating the date of Easter). They included zero (via the Latin word nulla meaning "none") as one of nineteen epacts, or the age of the moon on March 22. The first three epacts were nullae, xi, and xxii (written in minuscule or lower case). The first known computist to use zero was Dionysius Exiguus in 525. Only one instance of a Roman numeral for zero is known. About 725, Bede or one of his colleagues used the letter N, the initial of nullae, in a table of epacts, all written in Roman numerals.

Arabic Numbers - The Arabic numerals are the ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). They are descended from Indian numerals, and the Hindu-Arabic numeral system by which a sequence of digits such as "406" is read as a whole number was developed by Indian mathematicians. 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, and developed their 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 befitting their history, the digits (0,1,2,3,4,5,6,7,8,9) are also known as Hindu or Hindu-Arabic numerals. The reason that they are more commonly known as "Arabic numerals" in Europe and the Americas is that they were introduced to Europe in the tenth century from Arabs of North Africa. There they were (and still are) the digits used by western Arabs from Libya to Morocco. Arabs, on the other hand, call the system "Hindu numerals", referring to their origin in India. This term also includes the Eastern Arabic numerals used in the Mideast.

In English, the term Arabic numerals can be ambiguous. It most commonly refers to the numeral system widely used in Europe and the Americas. Arabic numerals is the conventional name for the entire family of related systems of Arabic and Indian numerals. It may also be intended to mean the numerals used by Arabs, in which case it generally refers to the Eastern Arabic numerals.

The decimal Hindu-Arabic numeral system was invented in India around 500 CE. The system was revolutionary in that it included a zero and positional notation. It is considered an important milestone in the development of mathematics. One may distinguish between this positional system, which is identical throughout the family, and the precise glyphs used to write the numerals, which vary regionally. The glyphs most commonly used in conjunction with the Latin alphabet since Early Modern times are 0 1 2 3 4 5 6 7 8 9.

Although the phrase "arabic numeral" is frequently capitalized, it is sometimes written in lower case, for instance in its entry in the Oxford English dictionary. This helps distinguish it from "Arabic numerals" as the East Arabic numerals specific to the Arabs.

The digits 1 to 9 in the Hindu-Arabic numeral system evolved from the Brahmi numerals. Buddhist inscriptions from around 300 BC use the symbols which became 1, 4 and 6. One century later, their use of the symbols which became 2, 7 and 9 was recorded. The first universally accepted inscription containing the use of the 0 glyph is first recorded in the 9th century, in an inscription at Gwalior in Central India dated to 870. By this time, the use of the glyph had already reached Persia, and was mentioned in Al-Khwarizmi's descriptions of Indian numerals. Numerous Indian documents on copper plates exist, with the same symbol for zero in them, dated back as far as the 6th century AD.

The numeral system came to be known to both the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals written about 825 in Arabic, and the Arab mathematician Al-Kindi, who wrote four volumes, "On the Use of the Indian Numerals" (Ketab fi Isti'mal al-'Adad al-Hindi) about 830. Their work was principally responsible for the diffusion of the Indian system of numeration in the Middle East and the West. In the 10th century, Middle-Eastern mathematicians extended the decimal numeral system to include fractions, as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952-53.

In the Arab world-until modern times-the Arabic numeral system was used only by mathematicians. Muslim scientists used the Babylonian numeral system, and merchants used the Abjad numerals. It was not until the Italian Fibonacci's early 13th century popularization that the Arabic numeral system was used by a large population outside India.

A distinctive West Arabic variant of the symbols begins to emerge around the 10th century in the Maghreb and Al-Andalus, called ghubar ("sand-table" or "dust-table") numerals, which is the direct ancestor to the modern Western Arabic numerals used throughout the world.

The first mentions of the numerals in the West are found in the Codex Vigilanus of 976. From the 980s, Gerbert of Aurillac (later, Pope Sylvester II) used his office to spread knowledge of the numerals in Europe. Gerbert studied in Barcelona in his youth. He was known to have requested mathematical treatises concerning the astrolabe from Lupitus of Barcelona after he had returned to France.