The scope of precision math for exact-value operations includes the exact-value data types (DECIMAL and integer types) and exact-value numeric literals. Approximate-value data types and numeric literals still are handled as floating-point numbers.

Exact-value numeric literals have an integer part or fractional part, or both. They may be signed. Examples: 1, .2, 3.4, -5, -6.78, +9.10.

Approximate-value numeric literals are represented in scientific notation with a mantissa and exponent. Either or both parts may be signed. Examples: 1.2E3, 1.2E-3, -1.2E3, -1.2E-3.

Numbers that look similar need not be both exact-value or both approximate-value. For example, 2.34 is an exact-value (fixed-point) number, whereas 2.34E0 is an approximate-value (floating-point) number.

The DECIMAL data type is a fixed-point type and calculations are exact. In MySQL, the DECIMAL type has several synonyms: NUMERIC, DEC, FIXED. The integer types also are exact-value types.

The FLOAT and DOUBLE data types are floating-point types and calculations are approximate. In MySQL, types that are synonymous with FLOAT or DOUBLE are DOUBLE PRECISION and REAL.

This section discusses the characteristics of the DECIMAL data type (and its synonyms) in MySQL 5.1, particularly with regard to the following:

Possible incompatibilities with applications that are written for older versions of MySQL are noted throughout this section.

The declaration syntax for a DECIMAL column is DECIMAL(M,D). The ranges of values for the arguments in MySQL 5.1 are as follows:

The maximum value of 65 for M means that calculations on DECIMAL values are accurate up to 65 digits. This limit of 65 digits of precision also applies to exact-value numeric literals, so the maximum range of such literals is different from before. (In older versions of MySQL, decimal values could have up to 254 digits. However, calculations were done using floating-point and thus were approximate, not exact.)

Values for DECIMAL columns in MySQL 5.1 are stored using a binary format that packs nine decimal digits into four bytes. The storage requirements for the integer and fractional parts of each value are determined separately. Each multiple of nine digits requires four bytes, and any digits left over require some fraction of four bytes. For example, a DECIMAL(18,9) column has nine digits on either side of the decimal point, so the integer part and the fractional part each require four bytes. A DECIMAL(20,10) column has ten digits on either side of the decimal point. Each part requires four bytes for nine of the digits, and one byte for the remaining digit.

The storage required for leftover digits is given by the following table:

Unlike some older versions of MySQL, DECIMAL columns in MySQL 5.1 do not store a leading + character or leading 0 digits. If you insert +0003.1 into a DECIMAL(5,1) column, it is stored as 3.1. Applications that rely on the older behavior must be modified to account for this change.

DECIMAL columns in MySQL 5.1 do not allow values larger than the range implied by the column definition. For example, a DECIMAL(3,0) column supports a range of -999 to 999. A DECIMAL(M,D) column allows at most M – D digits to the left of the decimal point. (This is not compatible with applications relying on older versions of MySQL that allowed storing an extra digit in lieu of a + sign.)

The SQL standard requires that the precision of NUMERIC(M,D) be exactly M digits. For DECIMAL(M,D), the standard requires a precision of at least M digits but allows more. In MySQL, DECIMAL(M,D) and NUMERIC(M,D) are the same, and both have a precision of exactly M digits.

For more detailed information about porting applications that relied on the old treatment of the DECIMAL data type, see the MySQL 5.0 Reference Manual.

With precision math, exact-value numbers are used as given whenever possible. For example, numbers in comparisons are used exactly as given without a change in value. In strict SQL mode, for INSERT into a column with an exact data type (DECIMAL or integer), a number is inserted with its exact value if it is within the column range. When retrieved, the value should be the same as what was inserted. (Without strict mode, truncation for INSERT is allowable.)

Handling of a numeric expression depends on what kind of values the expression contains:

If a numeric expression contains any strings, they are converted to double-precision floating-point values and the expression is approximate.

Inserts into numeric columns are affected by the SQL mode, which is controlled by the sql_mode system variable. (See Section 1.9.2, “Selecting SQL Modes”.) The following discussion mentions strict mode (selected by the STRICT_ALL_TABLES or STRICT_TRANS_TABLES mode values) and ERROR_FOR_DIVISION_BY_ZERO. To turn on all restrictions, you can simply use TRADITIONAL mode, which includes both strict mode and ERROR_FOR_DIVISION_BY_ZERO:

mysql> SET SQL_MODE='TRADITIONAL';

If a number is inserted into an exact type column (DECIMAL or integer), it should be inserted with its exact value if it is within the column range.

If the value has too many digits in the fractional part, rounding occurs and a warning is generated. Rounding is done as described in Rounding Behavior.

If the value has too many digits in the integer part, it is too large and is handled as follows:

Underflow is not detected, so underflow handing is undefined.

By default, division by zero produces a result of NULL and no warning. With the ERROR_FOR_DIVISION_BY_ZERO SQL mode enabled, MySQL handles division by zero differently:

In other words, inserts and updates involving expressions that perform division by zero can be treated as errors, but this requires ERROR_FOR_DIVISION_BY_ZERO in addition to strict mode.

Suppose that we have this statement:

INSERT INTO t SET i = 1/0;

This is what happens for combinations of strict and ERROR_FOR_DIVISION_BY_ZERO modes:

For inserts of strings into numeric columns, conversion from string to number is handled as follows if the string has non-numeric contents:

This section discusses precision math rounding for the ROUND() function and for inserts into DECIMAL columns.

The ROUND() function rounds differently depending on whether its argument is exact or approximate:

The following example shows how rounding differs for exact and approximate values:

mysql> SELECT ROUND(2.5), ROUND(25E-1);
+------------+--------------+
| ROUND(2.5) | ROUND(25E-1) |
+------------+--------------+
| 3          |            2 |
+------------+--------------+

For inserts into a DECIMAL column, the target is an exact data type, so rounding uses "round half up," regardless of whether the value to be inserted is exact or approximate:

mysql> CREATE TABLE t (d DECIMAL(10,0));
Query OK, 0 rows affected (0.00 sec)

mysql> INSERT INTO t VALUES(2.5),(2.5E0);
Query OK, 2 rows affected, 2 warnings (0.00 sec)
Records: 2  Duplicates: 0  Warnings: 2

mysql> SELECT d FROM t;
+------+
| d    |
+------+
| 3    |
| 3    |
+------+

This section provides some examples that show precision math query results in MySQL 5.1.

Example 1. Numbers are used with their exact value as given when possible:

mysql> SELECT .1 + .2 = .3;
+--------------+
| .1 + .2 = .3 |
+--------------+
|            1 |
+--------------+

However, for floating-point values, results are inexact:

mysql> SELECT .1E0 + .2E0 = .3E0;
+--------------------+
| .1E0 + .2E0 = .3E0 |
+--------------------+
|                  0 |
+--------------------+

Another way to see the difference in exact and approximate value handling is to add a small number to a sum many times. Consider the following stored procedure, which adds .0001 to a variable 1,000 times.

CREATE PROCEDURE p ()
BEGIN
  DECLARE i INT DEFAULT 0;
  DECLARE d DECIMAL(10,4) DEFAULT 0;
  DECLARE f FLOAT DEFAULT 0;
  WHILE i < 10000 DO
    SET d = d + .0001;
    SET f = f + .0001E0;
    SET i = i + 1;
  END WHILE;
  SELECT d, f;
END;

The sum for both d and f logically should be 1, but that is true only for the decimal calculation. The floating-point calculation introduces small errors:

+--------+------------------+
| d      | f                |
+--------+------------------+
| 1.0000 | 0.99999999999991 |
+--------+------------------+

Example 2. Multiplication is performed with the scale required by standard SQL. That is, for two numbers X1 and X2 that have scale S1 and S2, the scale of the result is S1 + S2:

mysql> SELECT .01 * .01;
+-----------+
| .01 * .01 |
+-----------+
| 0.0001    |
+-----------+

Example 3. Rounding behavior is well-defined:

In MySQL 5.1, rounding behavior (for example, with the ROUND() function) is independent of the implementation of the underlying C library, which means that results are consistent from platform to platform.

In MySQL 5.1, rounding for DECIMAL columns and exact-valued numbers uses the “round half up” rule. Values with a fractional part of .5 or greater are rounded away from zero to the nearest integer, as shown here:

mysql> SELECT ROUND(2.5), ROUND(-2.5);
+------------+-------------+
| ROUND(2.5) | ROUND(-2.5) |
+------------+-------------+
| 3          | -3          |
+------------+-------------+

However, rounding for floating-point values uses the C library, which on many systems uses the “round to nearest even” rule. Values with any fractional part on such systems are rounded to the nearest even integer:

mysql> SELECT ROUND(2.5E0), ROUND(-2.5E0);
+--------------+---------------+
| ROUND(2.5E0) | ROUND(-2.5E0) |
+--------------+---------------+
|            2 |            -2 |
+--------------+---------------+

Example 4. In strict mode, inserting a value that is too large results in overflow and causes an error, rather than truncation to a legal value.

When MySQL is not running in strict mode, truncation to a legal value occurs:

mysql> SET SQL_MODE='';
Query OK, 0 rows affected (0.00 sec)

mysql> CREATE TABLE t (i TINYINT);
Query OK, 0 rows affected (0.00 sec)

mysql> INSERT INTO t SET i = 128;
Query OK, 1 row affected, 1 warning (0.01 sec)

mysql> SELECT i FROM t;
+------+
| i    |
+------+
|  127 |
+------+
1 row in set (0.00 sec)

Howver, an overflow condition occurs if strict mode is in effect:

mysql> SET SQL_MODE='TRADITIONAL';
Query OK, 0 rows affected (0.00 sec)

mysql> CREATE TABLE t (i TINYINT);
Query OK, 0 rows affected (0.01 sec)

mysql> SET sql_mode='STRICT_ALL_TABLES';
Query OK, 0 rows affected (0.10 sec)

mysql> INSERT INTO t SET i = 128;
ERROR 1264 (22003): Out of range value adjusted for column 'i' at row 1

mysql> SELECT i FROM t;
Empty set (0.00 sec)

Example 5: In strict mode and with ERROR_FOR_DIVISION_BY_ZERO set, division by zero causes an error, and not a result of NULL.

In non-strict mode, division by zero has a result of NULL:

mysql> SET SQL_MODE='';
Query OK, 0 rows affected (0.00 sec)

mysql> CREATE TABLE t (i TINYINT);
Query OK, 0 rows affected (0.01 sec)

mysql> INSERT INTO t SET i = 1 / 0;
Query OK, 1 row affected (0.06 sec)

mysql> SELECT i FROM t;
+------+
| i    |
+------+
| NULL |
+------+
1 row in set (0.01 sec)

However, division by zero is an error if the proper SQL modes are in effect:

mysql> SET SQL_MODE='TRADITIONAL';
Query OK, 0 rows affected (0.00 sec)

mysql> CREATE TABLE t (i TINYINT);
Query OK, 0 rows affected (0.00 sec)

mysql> SET sql_mode='STRICT_ALL_TABLES,ERROR_FOR_DIVISION_BY_ZERO';
Query OK, 0 rows affected (0.00 sec)

mysql> INSERT INTO t SET i = 1 / 0;
ERROR 1365 (22012): Division by 0

mysql> SELECT i FROM t;
Empty set (0.01 sec)

Example 6. In MySQL 4 (before precision math was introduced), both exact-value and approximate-value literals were converted to double-precision floating-point values:

mysql> SELECT VERSION();
+-----------------+
| VERSION()       |
+-----------------+
| 4.0.25-standard |
+-----------------+
1 row in set (0.00 sec)


mysql> CREATE TABLE t SELECT 2.5 AS a, 25E-1 AS b;

mysql> DESCRIBE t;
+-------+-------------+------+-----+---------+-------+
| Field | Type        | Null | Key | Default | Extra |
+-------+-------------+------+-----+---------+-------+
| a     | double(3,1) |      |     | 0.0     |       |
| b     | double      |      |     | 0       |       |
+-------+-------------+------+-----+---------+-------+

In MySQL 5.1, the approximate-value literal still is converted to floating-point, but the exact-value literal is handled as DECIMAL:

mysql> SELECT VERSION();
+-----------------+
| VERSION()       |
+-----------------+
| 5.1.2-alpha-log |
+-----------------+
1 row in set (0.00 sec)

mysql> CREATE TABLE t SELECT 2.5 AS a, 25E-1 AS b;

mysql> DESCRIBE t;
+-------+--------------+------+-----+---------+-------+
| Field | Type         | Null | Key | Default | Extra |
+-------+--------------+------+-----+---------+-------+
| a     | decimal(2,1) | NO   |     | 0.0     |       |
| b     | double       | NO   |     | 0       |       |
+-------+--------------+------+-----+---------+-------+

Example 7. If the argument to an aggregate function is an exact numeric type, the result is also, with a scale at least that of the argument.

Consider these statements:

mysql> CREATE TABLE t (i INT, d DECIMAL, f FLOAT);
mysql> INSERT INTO t VALUES(1,1,1);
mysql> CREATE TABLE y SELECT AVG(i), AVG(d), AVG(f) FROM t;

Result in MySQL 4.0 or 4.1 (prior to the introduction of precision math in MySQL):

mysql> DESCRIBE y;
+--------+--------------+------+-----+---------+-------+
| Field  | Type         | Null | Key | Default | Extra |
+--------+--------------+------+-----+---------+-------+
| AVG(i) | double(17,4) | YES  |     | NULL    |       |
| AVG(d) | double(17,4) | YES  |     | NULL    |       |
| AVG(f) | double       | YES  |     | NULL    |       |
+--------+--------------+------+-----+---------+-------+

The result is a double no matter the argument type.

Result in MySQL 5.1:

mysql> DESCRIBE y;
+--------+---------------+------+-----+---------+-------+
| Field  | Type          | Null | Key | Default | Extra |
+--------+---------------+------+-----+---------+-------+
| AVG(i) | decimal(14,4) | YES  |     | NULL    |       |
| AVG(d) | decimal(14,4) | YES  |     | NULL    |       |
| AVG(f) | double        | YES  |     | NULL    |       |
+--------+---------------+------+-----+---------+-------+

The result is a double only for the floating-point argument. For exact type arguments, the result is also an exact type.