# 9.3. 數學函式及運算子

本節提供了 PostgreSQL 的數學運算方式。對於沒有標準數學約定的型別（例如，日期/時間型別），我們將在後續部分中介紹具體的行為。

[Table 9.4](#table-9-4-mathematical-operators) 列出了可用的數學運算子。

#### **Table 9.4. Mathematical Operators**

| Operator | Description                                      | Example      | Result |
| -------- | ------------------------------------------------ | ------------ | ------ |
| `+`      | addition                                         | `2 + 3`      | `5`    |
| `-`      | subtraction                                      | `2 - 3`      | `-1`   |
| `*`      | multiplication                                   | `2 * 3`      | `6`    |
| `/`      | division (integer division truncates the result) | `4 / 2`      | `2`    |
| `%`      | modulo (remainder)                               | `5 % 4`      | `1`    |
| `^`      | exponentiation (associates left to right)        | `2.0 ^ 3.0`  | `8`    |
| `\|/`    | square root                                      | `\|/ 25.0`   | `5`    |
| `\|\|/`  | cube root                                        | `\|\|/ 27.0` | `3`    |
| `!`      | factorial                                        | `5 !`        | `120`  |
| `!!`     | factorial (prefix operator)                      | `!! 5`       | `120`  |
| `@`      | absolute value                                   | `@ -5.0`     | `5`    |
| `&`      | bitwise AND                                      | `91 & 15`    | `11`   |
| `\|`     | bitwise OR                                       | `32 \| 3`    | `35`   |
| `#`      | bitwise XOR                                      | `17 # 5`     | `20`   |
| `~`      | bitwise NOT                                      | `~1`         | `-2`   |
| `<<`     | bitwise shift left                               | `1 << 4`     | `16`   |
| `>>`     | bitwise shift right                              | `8 >> 2`     | `2`    |

位元運算子僅適用於整數資料型別，也可用於位元字串型別的位元和位元變化，如 [Table 9.14](/the-sql-language/functions-and-operators/bit-string-functions-and-operators.md#table-9-14-bit-string-operators) 所示。

[Table 9.5](#table-9-5-mathematical-functions) 列出了可用的數學函數。在該表中，dp 表示雙精確度。這些函數中的許多函數都提供了多種形式，且具有不同的參數型別。除非另有說明，否則函數的任何形式都將回傳與其參數相同的資料型別。使用雙精確度資料的功能主要以主機系統的 C 函式庫實作； 因此，邊界情況下的準確性和行為可能會因主機系統而有所差異。

#### **Table 9.5. Mathematical Functions**

| Function                                                                                   | Return Type              | Description                                                                                                                                                                                                                                                                         | Example                                                                       | Result              |
| ------------------------------------------------------------------------------------------ | ------------------------ | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------- | ------------------- |
| `abs(`*`x`*)                                                                               | (same as input)          | absolute value                                                                                                                                                                                                                                                                      | `abs(-17.4)`                                                                  | `17.4`              |
| `cbrt(dp`)                                                                                 | `dp`                     | cube root                                                                                                                                                                                                                                                                           | `cbrt(27.0)`                                                                  | `3`                 |
| `ceil(dp` or `numeric`)                                                                    | (same as input)          | nearest integer greater than or equal to argument                                                                                                                                                                                                                                   | `ceil(-42.8)`                                                                 | `-42`               |
| `ceiling(dp` or `numeric`)                                                                 | (same as input)          | nearest integer greater than or equal to argument (same as `ceil`)                                                                                                                                                                                                                  | `ceiling(-95.3)`                                                              | `-95`               |
| `degrees(dp`)                                                                              | `dp`                     | radians to degrees                                                                                                                                                                                                                                                                  | `degrees(0.5)`                                                                | `28.6478897565412`  |
| `div(`*`y`* `numeric`, *`x`* `numeric`)                                                    | `numeric`                | integer quotient of *`y`*/*`x`*                                                                                                                                                                                                                                                     | `div(9,4)`                                                                    | `2`                 |
| `exp(dp` or `numeric`)                                                                     | (same as input)          | exponential                                                                                                                                                                                                                                                                         | `exp(1.0)`                                                                    | `2.71828182845905`  |
| `floor(dp` or `numeric`)                                                                   | (same as input)          | nearest integer less than or equal to argument                                                                                                                                                                                                                                      | `floor(-42.8)`                                                                | `-43`               |
| `ln(dp` or `numeric`)                                                                      | (same as input)          | natural logarithm                                                                                                                                                                                                                                                                   | `ln(2.0)`                                                                     | `0.693147180559945` |
| `log(dp` or `numeric`)                                                                     | (same as input)          | base 10 logarithm                                                                                                                                                                                                                                                                   | `log(100.0)`                                                                  | `2`                 |
| `log10(dp` or `numeric`)                                                                   | (same as input)          | base 10 logarithm                                                                                                                                                                                                                                                                   | `log10(100.0)`                                                                | `2`                 |
| `log(`*`b`* `numeric`, *`x`* `numeric`)                                                    | `numeric`                | logarithm to base *`b`*                                                                                                                                                                                                                                                             | `log(2.0, 64.0)`                                                              | `6.0000000000`      |
| `mod(`*`y`*, *`x`*)                                                                        | (same as argument types) | remainder of *`y`*/*`x`*                                                                                                                                                                                                                                                            | `mod(9,4)`                                                                    | `1`                 |
| `pi()`                                                                                     | `dp`                     | “π” constant                                                                                                                                                                                                                                                                        | `pi()`                                                                        | `3.14159265358979`  |
| `power(`*`a`* `dp`, *`b`* `dp`)                                                            | `dp`                     | *`a`* raised to the power of *`b`*                                                                                                                                                                                                                                                  | `power(9.0, 3.0)`                                                             | `729`               |
| `power(`*`a`* `numeric`, *`b`* `numeric`)                                                  | `numeric`                | *`a`* raised to the power of *`b`*                                                                                                                                                                                                                                                  | `power(9.0, 3.0)`                                                             | `729`               |
| `radians(dp`)                                                                              | `dp`                     | degrees to radians                                                                                                                                                                                                                                                                  | `radians(45.0)`                                                               | `0.785398163397448` |
| `round(dp` or `numeric`)                                                                   | (same as input)          | round to nearest integer                                                                                                                                                                                                                                                            | `round(42.4)`                                                                 | `42`                |
| `round(`*`v`* `numeric`, *`s`* `int`)                                                      | `numeric`                | round to *`s`* decimal places                                                                                                                                                                                                                                                       | `round(42.4382, 2)`                                                           | `42.44`             |
| `scale(numeric`)                                                                           | `integer`                | scale of the argument (the number of decimal digits in the fractional part)                                                                                                                                                                                                         | `scale(8.41)`                                                                 | `2`                 |
| `sign(dp` or `numeric`)                                                                    | (same as input)          | sign of the argument (-1, 0, +1)                                                                                                                                                                                                                                                    | `sign(-8.4)`                                                                  | `-1`                |
| `sqrt(dp` or `numeric`)                                                                    | (same as input)          | square root                                                                                                                                                                                                                                                                         | `sqrt(2.0)`                                                                   | `1.4142135623731`   |
| `trunc(dp` or `numeric`)                                                                   | (same as input)          | truncate toward zero                                                                                                                                                                                                                                                                | `trunc(42.8)`                                                                 | `42`                |
| `trunc(`*`v`* `numeric`, *`s`* `int`)                                                      | `numeric`                | truncate to *`s`* decimal places                                                                                                                                                                                                                                                    | `trunc(42.4382, 2)`                                                           | `42.43`             |
| `width_bucket(`*`operand`* `dp`, *`b1`* `dp`, *`b2`* `dp`, *`count`* `int`)                | `int`                    | return the bucket number to which *`operand`* would be assigned in a histogram having *`count`* equal-width buckets spanning the range *`b1`* to *`b2`*; returns `0` or *`count`*+1 for an input outside the range                                                                  | `width_bucket(5.35, 0.024, 10.06, 5)`                                         | `3`                 |
| `width_bucket(`*`operand`* `numeric`, *`b1`* `numeric`, *`b2`* `numeric`, *`count`* `int`) | `int`                    | return the bucket number to which *`operand`* would be assigned in a histogram having *`count`* equal-width buckets spanning the range *`b1`* to *`b2`*; returns `0` or *`count`*+1 for an input outside the range                                                                  | `width_bucket(5.35, 0.024, 10.06, 5)`                                         | `3`                 |
| `width_bucket(`*`operand`* `anyelement`, *`thresholds`* `anyarray`)                        | `int`                    | return the bucket number to which *`operand`* would be assigned given an array listing the lower bounds of the buckets; returns `0` for an input less than the first lower bound; the *`thresholds`* array *must be sorted*, smallest first, or unexpected results will be obtained | `width_bucket(now(), array['yesterday', 'today', 'tomorrow']::timestamptz[])` | `2`                 |

[Table 9.6](https://www.postgresql.org/docs/12/functions-math.html#FUNCTIONS-MATH-RANDOM-TABLE) shows functions for generating random numbers.

#### **Table 9.6. Random Functions**

| Function      | Return Type | Description                                                                      |
| ------------- | ----------- | -------------------------------------------------------------------------------- |
| `random()`    | `dp`        | random value in the range 0.0 <= x < 1.0                                         |
| `setseed(dp`) | `void`      | set seed for subsequent `random()` calls (value between -1.0 and 1.0, inclusive) |

The `random()` function uses a simple linear congruential algorithm. It is fast but not suitable for cryptographic applications; see the [pgcrypto](https://www.postgresql.org/docs/12/pgcrypto.html) module for a more secure alternative. If `setseed()` is called, the results of subsequent `random()` calls in the current session are repeatable by re-issuing `setseed()` with the same argument.

[Table 9.7](https://www.postgresql.org/docs/12/functions-math.html#FUNCTIONS-MATH-TRIG-TABLE) shows the available trigonometric functions. All these functions take arguments and return values of type `double precision`. Each of the trigonometric functions comes in two variants, one that measures angles in radians and one that measures angles in degrees.

#### **Table 9.7. Trigonometric Functions**

| Function (radians)    | Function (degrees)     | Description                    |
| --------------------- | ---------------------- | ------------------------------ |
| `acos(`*`x`*)         | `acosd(`*`x`*)         | inverse cosine                 |
| `asin(`*`x`*)         | `asind(`*`x`*)         | inverse sine                   |
| `atan(`*`x`*)         | `atand(`*`x`*)         | inverse tangent                |
| `atan2(`*`y`*, *`x`*) | `atan2d(`*`y`*, *`x`*) | inverse tangent of *`y`*/*`x`* |
| `cos(`*`x`*)          | `cosd(`*`x`*)          | cosine                         |
| `cot(`*`x`*)          | `cotd(`*`x`*)          | cotangent                      |
| `sin(`*`x`*)          | `sind(`*`x`*)          | sine                           |
| `tan(`*`x`*)          | `tand(`*`x`*)          | tangent                        |

#### Note

Another way to work with angles measured in degrees is to use the unit transformation functions `radians()` and `degrees()` shown earlier. However, using the degree-based trigonometric functions is preferred, as that way avoids round-off error for special cases such as `sind(30)`.

[Table 9.8](https://www.postgresql.org/docs/12/functions-math.html#FUNCTIONS-MATH-HYP-TABLE) shows the available hyperbolic functions. All these functions take arguments and return values of type `double precision`.

#### **Table 9.8. Hyperbolic Functions**

| Function       | Description                | Example    | Result |
| -------------- | -------------------------- | ---------- | ------ |
| `sinh(`*`x`*)  | hyperbolic sine            | `sinh(0)`  | `0`    |
| `cosh(`*`x`*)  | hyperbolic cosine          | `cosh(0)`  | `1`    |
| `tanh(`*`x`*)  | hyperbolic tangent         | `tanh(0)`  | `0`    |
| `asinh(`*`x`*) | inverse hyperbolic sine    | `asinh(0)` | `0`    |
| `acosh(`*`x`*) | inverse hyperbolic cosine  | `acosh(1)` | `0`    |
| `atanh(`*`x`*) | inverse hyperbolic tangent | `atanh(0)` | `0`    |


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://docs.postgresql.tw/the-sql-language/functions-and-operators/mathematical-functions-and-operators.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.