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Generalising the positional system to infinite sequences of digits yields an intuitive description of the real line.
Decimal systemIn the decimal (base-10) Hindu-Arabic numeral system, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on. Fractional values are indicated by a separator, which varies by locale. Usually this separator is a period or full stop, or a comma. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10-1 (0.1), the second position 10-2 (0.01), and so on for each successive position.
or
Digits and numeralsFor a positional system up to ten the ubiquitous digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used, for octal only eight digits up to 7 and for binary only two digits 0 and 1 are needed. For bases above 10, extra digits are needed. For hexadecimal the first six letters of the alphabet A, B, C, D, E, and F are commonly used for decimal values 10 to 15. The alphabet can cover numeral systems with a base up to 10 + 26 = 36. However, some uppercase letters can be confused with 'existing' digits such as an I with a 1 and O with 0. When these are omitted it can reach 34. Adding lowercase letters (none of them can be confused with 'existing' digits, except l in some fonts) extends the digit set to 62 (or 60 when uppercase I and O are omitted). For a base 60 system a 'mixed' base with 10 as 'secondary' base is commonly used, please see below. Sexagesimal systemThe sexagesimal or base sixty system was used for the integral and fractional portions of Babylonian numerals, by Hellenistic astronomers using Greek numerals for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written 10°25′59″23‴31''''12''''' or 10°25I59II23III31IV12V. Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz', which be useful for use in URLs, etc., but it is not very intelligible to humans. In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees. Non-positional positionsEach position does not need to be positional itself. Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a zero symbol), whereas Babylonian numerals used groups of two kinds of wedges representing ones and tens (a narrow vertical wedge ( | ) and an open left pointing wedge (<)) — up to 14 symbols per position (5 tens (<<<<<) and 9 ones ( ||||||||| ) grouped into one or two near squares containing up to three tiers of symbols, or a place holder (\\) for the lack of a position). A hypothetical Roman numeral positional system would separate each position with punctuation marks but would not necessarily require a symbol for zero. For example, 144 might be I.IV.IV. in decimal notation (medieval Roman numerals were always terminated by a point to show that they were a number). To indicate zero, its position might not be present, for example I.IV.. would mean 140. About 725, Bede or a colleague used N for zero (the initial of the Latin word nulla meaning nothing), so the latter might be I.IV.N. See also
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