Positional notation

From Academic Kids

Positional notation is a numeral system in which each position is related to the next by a constant multiplier called the base of that numeral system. Each position may be represented by a unique symbol or by a limited set of symbols. The resultant value of each position is the value of its symbol or symbols multiplied by a power of the base. The total value of a positional number is the total of the resultant values of all positions. The decimal system uses ten unique symbols, whereas the sexagesimal system usually uses a pseudo-decimal system for each position and separates each position from the next by punctuation. Modern computers use binary, octal, and hexadecimal numbers, the latter using decimal numerals (0–9) plus the letters A–F to provide the sixteen possible symbols in each position.

The idea of indicating magnitude by means of position was embodied long ago by the use of the abacus in all its various forms. With an abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system such as Roman Numerals. This approach required no memorization of tables (as does positional notation) and could produce results for all practical purposes very quickly. For four centuries (13th - 16th) there was strong disagreement between those who believed in adopting the positional system and those who wanted to stay with the additive-system-plus-abacus. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern bank checks require a natural language spelling of an amount, as well as the amount itself, to prevent such fraud. The abacus was in widespread use in Japan and other Asian countries until very recent times, when it was replaced by calculators.

The real value of positional notation turned out to be its ability to invite the further study of numbers. Integers, rational numbers, and place-holders (e.g. zero) were long known about, but irrational numbers, infinity, transfinite numbers, and imaginary numbers were all concepts that could only be discovered once the idea of a continuous number line was implied by positional notation.

Contents

Decimal system

In the decimal or base 10 number system, each position starting from the right is a higher power of 10. The first position represents 100, the second position 101, the third 102, the fourth 103, 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, the second position 10-2, and so on for each successive position.

As an example, the number 2674 in a base 10 number system is :

( 2 × 103 ) + ( 6 × 102 ) + ( 7 × 101 ) + ( 4 × 100 )

or

( 2 × 1000 ) + ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 )

Sexagesimal system

The 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 1025'59" (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 1025'59.392", they would have written 1025'59''23'''31''''12''''' or 1025I59II23III31IV12V.

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.

Non-positional positions

Each 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.

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