Arithmetic precision

The precision of a value describes the number of digits that are used to express that value. In Mathematics and Computer science, a digit is a symbol (a number symbol e In a scientific setting this would be the total number of digits (sometimes called the significant figures or significant digits) or, less commonly, the number of fractional digits or decimal places (the number of digits following the point). The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy In Mathematics and Computing, a Radix point (or radix character) is the symbol used in numerical representations to separate the Integer This second definition is useful in financial and engineering applications where the number of digits in the fractional part has particular importance.

In both cases, the term precision can be used to describe the position at which an inexact result will be rounded. For example, in floating point arithmetic, a result is rounded to a given or fixed precision, which is the length of the resulting significand. In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. The significand (also Coefficient or Mantissa) is the part of a floating-point number that contains its significant digits In financial calculations, a number is often rounded to a given number of places (for example, to two places after the decimal separator for many world currencies). In a positional Numeral system, the decimal separator is a Symbol used to mark the boundary between the integral and the fractional

As an illustration, the decimal quantity 12. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. 345 can be expressed with various numbers of significant digits or decimal places. If insufficient precision is available then the number is rounded in some manner to fit the available precision. For lip-rounding in phonetics see Labialisation and Roundedness. The following table shows the results for various total precisions and decimal places, with the results rounded to nearest where ties round up or to an even digit (the most common rounding modes).

Note that it is often not appropriate to display a figure with more digits than that which can be measured. For instance, if a device measures to the nearest gram and gives a reading of 12. For other uses of the words gram or gramme see Gram (disambiguation. 345 kg, it would create false precision if you were to express this measurement as 12. False precision occurs when numerical data are presented in a manner that implies better precision than is actually the case since precision is a limit to accuracy this often 34500 kg.

Precision	Rounded to significant digits	Rounded to decimal places
Five	12. 345	12. 34500
Four	12. 35	12. 3450
Three	12. 3	12. 345
Two	12	12. 35
One	1 × 10¹	12. 3
Zero	n/a	12

The representation of a positive number x to a precision of p significant digits has a numerical value that is given by the formula

round(10⁻ⁿ·x)·10ⁿ, where n = floor(log₁₀ x) + 1 – p. For lip-rounding in phonetics see Labialisation and Roundedness. In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The

For a negative number, the numerical value is minus that of the absolute value. The number 0, to any precision, can be taken to be 0.

See also