Hittites Base 10 Unit Fractions

Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands typically having ten fingers/digits.

Base 10 units, also known as multibase arithmetic blocks (MAB) or Dienes blocks (after their creator, mathematician and educationalist Zoltán Pál Dienes), are a mathematical manipulative used by students to learn basic mathematical concepts including addition, subtraction, number sense, place value and counting.

Simply put, a unit fraction is a rational number written as a fraction, where, the numerator is one and the denominator is a positive integer. A unit fraction is, therefore, the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc…

Some non-mathematical ancient texts such as the Vedas, dating back to (1700–900 BC) make use of decimals and mathematical decimal fractions. Hittite hieroglyphs were also strictly decimal.

The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000.

J. Lennart Berggren, a Canadian science historian, and author of "The Mathematics of Egypt, Mesopotamia, China, India, and Islam", denotes that positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century AD. The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, a Flemish mathematician, physicist and military engineer, but did not develop any notation to represent them.

The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century AD. Al Khwarizmi, a Persian astronomer and mathematician,?introduced fraction to Islamic countries in the early 9th century AD; a Chinese authors have alleged that his fraction presentation was an exact copy of traditional Chinese mathematical fraction from Sunzi Suanjing, a mathematical treatise written during 3rd to 5th centuries AD. The written Chinese decimal fractions were non-positional, while, counting rod fractions were positional.

The study of Indian math, be it zero, infinity, and base 10 cannot be complete without the study of its diffusion, and receipt of diffusion from nearby cultures, according to Milo Gardner, and American Code Breaker and Cryptanalyst. As another under-reported subject, Hittites used base 10 unit fractions, not base 60 fractions as one would expect being closely identified with Anatolia and Babylonian law.

Where did the Hittite unit fractions, covered by George Ifrah, a French professor of mathematics, author and a self-taught historian of mathematics, especially numerals; in his latest book "The Universal?History of Numbers", originate? Were Indian mathematicians and traders first to develop unit fractions? If so, why did Sumer and Babylon use base 60, if they too received large segments of its early math from India?

Clearly India, by 1500 BC was closely aligned with the Babylonian 'water clocks, time keeping and other important science, yet India used base 10, without unit fractions. The question concerning which developed what when, may be open in minor respects. However, after 1500 BC Indian mathematicians did use base 10 unit fractions, as aside subject, for puzzle and game solving, much as we do today in the USA and Europe. Could the Indians have used unit fractions earlier, and provided them to the Hittites (and Minoans)?

Stated another way, very early, if not throughout its history, Hittites used base 10 unit fractions, vividly looking like Egyptian fractions. The same can be said of Minoans. The Mediterranean trading region, often dominated by Egypt required the use of base 10 unit fractions, and not base 10 from India, correct?

Why then have historians stressed the Babylonian law side of Hittites, a cultural fact that was not true in several important mathematical respects?

Could be that an open discussion of diffusion, taking into account India, zero, base 10 or base 60 infinite numeration systems is very difficult to pin down, time and place wise, with India being its originator?

However, the finite side of base 10 unit fractions is rather easy to pin down. Rather than stressing the objective side of ancient mathematical documents, as the RMP, EMLR, Moscow P, Hibeh P. and closely aligned Hittite and Minoan base 10 unit fractions, discussions often ranging into fuzzy myths. Is myth being used as a strawman to confuse the scientific aspects of vivid cultural diffusions, placing Babylonians and/or Indians into a higher role in developing Western mathematics than the two cultures actually produced?

Again, the under reporting of the Indian Valley may one day show the date(s) that base 10 unit fractions first appeared there. Until that data appears, and confirms predating of Egyptian trade and intellectual use of unit fractions, the hard facts of several Egyptian texts should be considered a primary subject soft study (and not reduced to a scrap heal - as Otto E. Neugebauer, an Austrian-American mathematician and historian of science, discusses the sign of intellectual decline in the "Exact Sciences of Antiquity".

Finally, Hittites and Minoans used base 10 unit fractions, probably stated only in an infinite series context. However, in these two cases, one day the finite series of Egyptian hieratic numeration may one day be found, totally changing the view of the role of Egyptian math in the regions...

Food for thought!