Everything about Arithmetic Precision totally explained
The
precision of a value describes the number of
digits that are used to express that value. 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). 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 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).
As an illustration, the
decimal quantity 12.345 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. 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's 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 12.345
kg, it would create
false precision if you were to express this measurement as 12.34500 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's given by the formula
» round(10
−n·
x)·10
n, where
n =
floor(log
10 x) + 1 –
p.
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.
Further Information
Get more info on 'Arithmetic Precision'.
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