Arithmetic operation rules: basic operators

The basic operators (addition, subtraction, multiplication, and division) operate on numbers as follows.

Addition and subtraction

If either number is 0, the other number, rounded to NUMERIC DIGITS digits, if necessary, is used as the result (with sign adjustment as appropriate). Otherwise, the two numbers are extended on the right and left as necessary, up to a total maximum of DIGITS + 1 digits (the number with the smaller absolute value may, therefore, lose some or all of its digits on the right) and are then added or subtracted as appropriate.

Example

xxx.xxx  +  yy.yyyyy

becomes:


   xxx.xxx00
+  0yy.yyyyy
-------------
   zzz.zzzzz

The result is then rounded to the current setting of NUMERIC DIGITS if necessary (taking into account any extra carry digit on the left after addition, but otherwise counting from the position corresponding to the most significant digit of the terms being added or subtracted). Finally, any insignificant leading zeros are removed.

The prefix operators are evaluated using the same rules; the operations +number and -number are calculated as 0+number and 0-number, respectively.

Multiplication

The numbers are multiplied together (long multiplication) resulting in a number that can be as long as the sum of the lengths of the two operands.

Example

             xxx.xxx * yy.yyyyy

becomes:

             zzzzz.zzzzzzzz

The result is then rounded, counting from the first significant digit of the result, to the current setting of NUMERIC DIGITS.

Division

For the following division, the following steps are taken:

yyy / xxxxx

First the number yyy is extended with zeros on the right until it is larger than the number xxxxx (with note being taken of the change in the power of ten that this implies). Thus, in this example, yyy might become yyy00. Traditional long division then takes place. This might be written:

              zzzz
           +---------
    xxxxx  | yyy00

The length of the result (zzzz) is such that the rightmost z is at least as far right as the rightmost digit of the (extended) y number in the example. During the division, the y number is extended further as necessary. The z number may increase up to NUMERIC DIGITS+1 digits, at which point the division stops and the result is rounded. Following completion of the division (and rounding if necessary), insignificant trailing zeros are removed.

Examples of basic operators

The following examples illustrate the main implications of the rules just described.

/* With:  Numeric digits 5 */
12+7.00     ->    19.00
1.3-1.07    ->     0.23
1.3-2.07    ->    -0.77
1.20*3      ->     3.60
7*3         ->    21
0.9*0.8     ->     0.72
1/3         ->     0.33333
2/3         ->     0.66667
5/2         ->     2.5
1/10        ->     0.1
12/12       ->     1
8.0/2       ->     4

Note: With all the basic operators, the position of the decimal point in the terms being operated upon is arbitrary. The operations may be carried out as integer operations with the exponent being calculated and applied afterward. Therefore, the significant digits of a result are not in any way dependent on the position of the decimal point in either of the terms involved in the operation.