The residents of the island of Mangareva in French Polynesia were using a hybrid binary- decimal system before 1450. The numerical value is obtained by adding one to the sum of place values. Four short syllables "0000" is the first pattern and corresponds to the value one. In Pingala's system, the numbers start from number one, and not zero. The binary representations in Pingala's system increases towards the right, and not to the left like in the binary numbers of the modern positional notation. "Chandaḥśāstra" literally translates to science of meters in Sanskrit. Pingala's Hindu classic titled Chandaḥśāstra (8.23) describes the formation of a matrix in order to give a unique value to each meter. They were known as laghu (light) and guru (heavy) syllables.
He used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code. 2nd century BC) developed a binary system for describing prosody. Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. The Song Dynasty scholar Shao Yong (1011–1077) rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Eight trigrams (Bagua) and a set of 64 hexagrams ("sixty-four" gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. It is based on taoistic duality of yin and yang. The binary notation in the I Ching is used to interpret its quaternary divination technique. The I Ching dates from the 9th century BC in China. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC. In this method, multiplying one number by a second is performed by a sequence of steps in which a value (initially the first of the two numbers) is either doubled or has the first number added back into it the order in which these steps are to be performed is given by the binary representation of the second number. The method used for ancient Egyptian multiplication is also closely related to binary numbers.
Early forms of this system can be found in documents from the Fifth Dynasty of Egypt, approximately 2400 BC, and its fully developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt, approximately 1200 BC. Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions (not related to the binary number system) and Horus-Eye fractions (so called because many historians of mathematics believe that the symbols used for this system could be arranged to form the eye of Horus, although this has been disputed). Leibniz was specifically inspired by the Chinese I Ching.Īrithmetic values thought to have been represented by parts of the Eye of Horus However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, and Gottfried Leibniz. 7 Conversion to and from other numeral systems.Signed-digit representation ( balanced ternary).