A variable of the type decimal can hold decimal numbers.
The numbers are floating point. This means that the fraction (mantissa) and exponent are separate.
The numbers can be negative and positive.
A decimal variable can hold each whole value between +/-1 quadrillion (-10^15). Above 1 quadrillion and below one, it can hold 15 decimal digits and an exponent of two digits.
This table defines the minimum and maximum values:
sxx is the current exponent, one sign and two digits. For example, if the current variable value is
0.1 E+10, then the smallest change is
0.000 000 000 000 001 E+10.
|Special Value||Can be|
|Not a Number (||No|
|Undefined or Null||No|
In summary, a decimal variable can only hold numbers.
-999 999 999 999 999 E+00to
+999 999 999 999 999 E+00.
-999 999 999 999 999 E+XXto
+999 999 999 999 999 E+XXfor XX from 0 to -15. The exponent negated is the number of digits after the decimal point.
All computations must be done using a certain precision. The computations must give the same result if the decimal type has exactly 15 digits precision and if it has any higher precision. In other words, do not assume that the precision is, for example, 16 or 31 digits. If a computation gives an answer with 15 digits, it must produce the same result with 17 or 100 digits of precision.
The Progsbase system may be able to detect dependencies on the precision by
Computing the nth decimal digit of some operators might require a high number of iterations for some values. Thus, such computations belong in specialized libraries.
If the actual calculations on the decimal variable type is done using double precision IEEE Standard for Floating-Point Arithmetic (IEEE 754). Then, the following calculation would result in
y = true, but will result in
y = false if using 100 decimal digit precision.
x = 1 / 10 x = x * 10 x = x - 1 x = x * 1000000000 x = x * 100000000 x = round(x) y = x == 6
The following calculation works as long as the precision of the calculations used are at least 15 decimal digits floating point with three decimal digit exponent. This will always result in
y = true as long as the requirements of the decimal variable type are met.
x = 1 / 10 epsilon = 0.00001 y = |x - 0.1| < epsilon
This is known as an epsilon comparison.
An important note about finite precision (including binary and decimal) floating point numbers is that they are an approximation of the actual value you are storing. Hence, they must always be rounded before considered. e.g. when storing
34.65 / 10 in a double, you are actually storing
3.464999999999999857891452847979962825775146484375, which must first be rounded to 15 decimal digits before it can be considered, i.e. it must be rounded to
3.46500000000000. Hence, if we now are to round to 2 decimal digits, to get a currency amount, we get the correct
The same problems happens with decimal floating points:
3.465/27*27 = 3.464999999999991 which if rounded directly to two digits gives
3.46. The correct way is to round once to get the actual number:
3.465000000, and then again to get the rounded number:
The following is an elaboration with examples. The parts of the process are:
First, decide on a precision that the final calculated value will have. This depends on
For example, we decide on 10 decimal digits.
The field of mathematics with the theory for this is called the Calculus of Errors.
For more information, see arithmetic expressions.
The value you give to the program. Almost exclusively, numbers are written as decimal, even when stored as binary.
34.65 / 10
3.465 / 27 * 27
The value stored, e.g. in memory or on disk.
After calculations have been done on the stored value, we want to read out the value to use it. The first stage is to calculate the actual value. This means that we round to the precision we have determined is sufficient. For the example we are following, we determined that 10 digits was sufficient.
The actual value is stored in the same number system as is was entered, in this example decimal.
After the actual value has been determined, we can do further calculations if necessary. In this example, we are calculating with money, so we want to round so we get two digits after the decimal point.
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