Positional notation

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.

Related Topics:
Numeral system - Constant - Decimal - Sexagesimal - Binary - Octal - Hexadecimal

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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.

Related Topics:
Abacus - Roman Numerals

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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.

Related Topics:
Integer - Rational number - Irrational number - Infinity - Transfinite number - Imaginary number

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