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Areal data type is a data type used in a computer program to represent an approximation of a real number. Because the real numbers are not countable, computers cannot represent them exactly using a finite amount of information. Most often, a computer will use a rational approximation to a real number.

The most general data type for a rational number (a number that can be expressed as a fraction) stores the numerator and the denominator as integers. For example 1/3, which can be calculated to any desired precision. Rational number are used, for example, in Interpress from Xerox Corporation.[1]

A floating-point data type is a compromise between the flexibility of a general rational number data type and the speed of fixed-point arithmetic. It uses some of the bits in the data type to specify an exponent for the denominator, today usually power of two although both ten and sixteen have been used.[2]

involving subexpressions y and theta1 through thetaN (including the case where N is zero) will be well formed if and only if the corresponding assignment statement is well-formed. For densities allowing real y values, the log probability density function is used,

The syntax of constants for the numeric types is described in Section 4.1.2. The numeric types have a full set of corresponding arithmetic operators and functions. Refer to Chapter 9 for more information. The following sections describe the types in detail.

The types smallint, integer, and bigint store whole numbers, that is, numbers without fractional components, of various ranges. Attempts to store values outside of the allowed range will result in an error.

The type integer is the common choice, as it offers the best balance between range, storage size, and performance. The smallint type is generally only used if disk space is at a premium. The bigint type is designed to be used when the range of the integer type is insufficient.

The type numeric can store numbers with a very large number of digits. It is especially recommended for storing monetary amounts and other quantities where exactness is required. Calculations with numeric values yield exact results where possible, e.g., addition, subtraction, multiplication. However, calculations on numeric values are very slow compared to the integer types, or to the floating-point types described in the next section.

We use the following terms below: The precision of a numeric is the total count of significant digits in the whole number, that is, the number of digits to both sides of the decimal point. The scale of a numeric is the count of decimal digits in the fractional part, to the right of the decimal point. So the number 23.5141 has a precision of 6 and a scale of 4. Integers can be considered to have a scale of zero.

If the scale of a value to be stored is greater than the declared scale of the column, the system will round the value to the specified number of fractional digits. Then, if the number of digits to the left of the decimal point exceeds the declared precision minus the declared scale, an error is raised. For example, a column declared as

Beginning in PostgreSQL 15, it is allowed to declare a numeric column with a negative scale. Then values will be rounded to the left of the decimal point. The precision still represents the maximum number of non-rounded digits. Thus, a column declared as

will round values to the nearest thousand and can store values between -99000 and 99000, inclusive. It is also allowed to declare a scale larger than the declared precision. Such a column can only hold fractional values, and it requires the number of zero digits just to the right of the decimal point to be at least the declared scale minus the declared precision. For example, a column declared as

PostgreSQL permits the scale in a numeric type declaration to be any value in the range -1000 to 1000. However, the SQL standard requires the scale to be in the range 0 to precision. Using scales outside that range may not be portable to other database systems.

Numeric values are physically stored without any extra leading or trailing zeroes. Thus, the declared precision and scale of a column are maximums, not fixed allocations. (In this sense the numeric type is more akin to varchar(n) than to char(n).) The actual storage requirement is two bytes for each group of four decimal digits, plus three to eight bytes overhead.

The infinity values behave as per mathematical expectations. For example, Infinity plus any finite value equals Infinity, as does Infinity plus Infinity; but Infinity minus Infinity yields NaN (not a number), because it has no well-defined interpretation. Note that an infinity can only be stored in an unconstrained numeric column, because it notionally exceeds any finite precision limit.

The NaN (not a number) value is used to represent undefined calculational results. In general, any operation with a NaN input yields another NaN. The only exception is when the operation's other inputs are such that the same output would be obtained if the NaN were to be replaced by any finite or infinite numeric value; then, that output value is used for NaN too. (An example of this principle is that NaN raised to the zero power yields one.)

The data types real and double precision are inexact, variable-precision numeric types. On all currently supported platforms, these types are implementations of IEEE Standard 754 for Binary Floating-Point Arithmetic (single and double precision, respectively), to the extent that the underlying processor, operating system, and compiler support it.

Inexact means that some values cannot be converted exactly to the internal format and are stored as approximations, so that storing and retrieving a value might show slight discrepancies. Managing these errors and how they propagate through calculations is the subject of an entire branch of mathematics and computer science and will not be discussed here, except for the following points:

If you want to do complicated calculations with these types for anything important, especially if you rely on certain behavior in boundary cases (infinity, underflow), you should evaluate the implementation carefully.

On all currently supported platforms, the real type has a range of around 1E-37 to 1E+37 with a precision of at least 6 decimal digits. The double precision type has a range of around 1E-307 to 1E+308 with a precision of at least 15 digits. Values that are too large or too small will cause an error. Rounding might take place if the precision of an input number is too high. Numbers too close to zero that are not representable as distinct from zero will cause an underflow error.

By default, floating point values are output in text form in their shortest precise decimal representation; the decimal value produced is closer to the true stored binary value than to any other value representable in the same binary precision. (However, the output value is currently never exactly midway between two representable values, in order to avoid a widespread bug where input routines do not properly respect the round-to-nearest-even rule.) This value will use at most 17 significant decimal digits for float8 values, and at most 9 digits for float4 values.

For compatibility with output generated by older versions of PostgreSQL, and to allow the output precision to be reduced, the extra_float_digits parameter can be used to select rounded decimal output instead. Setting a value of 0 restores the previous default of rounding the value to 6 (for float4) or 15 (for float8) significant decimal digits. Setting a negative value reduces the number of digits further; for example -2 would round output to 4 or 13 digits respectively.

IEEE 754 specifies that NaN should not compare equal to any other floating-point value (including NaN). In order to allow floating-point values to be sorted and used in tree-based indexes, PostgreSQL treats NaN values as equal, and greater than all non-NaN values.

PostgreSQL also supports the SQL-standard notations float and float(p) for specifying inexact numeric types. Here, p specifies the minimum acceptable precision in binary digits. PostgreSQL accepts float(1) to float(24) as selecting the real type, while float(25) to float(53) select double precision. Values of p outside the allowed range draw an error. float with no precision specified is taken to mean double precision.

The data types smallserial, serial and bigserial are not true types, but merely a notational convenience for creating unique identifier columns (similar to the AUTO_INCREMENT property supported by some other databases). In the current implementation, specifying:

Because smallserial, serial and bigserial are implemented using sequences, there may be "holes" or gaps in the sequence of values which appears in the column, even if no rows are ever deleted. A value allocated from the sequence is still "used up" even if a row containing that value is never successfully inserted into the table column. This may happen, for example, if the inserting transaction rolls back. See nextval() in Section 9.17 for details.

To insert the next value of the sequence into the serial column, specify that the serial column should be assigned its default value. This can be done either by excluding the column from the list of columns in the INSERT statement, or through the use of the DEFAULT key word.

The type names serial and serial4 are equivalent: both create integer columns. The type names bigserial and serial8 work the same way, except that they create a bigint column. bigserial should be used if you anticipate the use of more than 231 identifiers over the lifetime of the table. The type names smallserial and serial2 also work the same way, except that they create a smallint column.

The sequence created for a serial column is automatically dropped when the owning column is dropped. You can drop the sequence without dropping the column, but this will force removal of the column default expression.

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