# Field (mathematics) > 실수와 동일한 방식으로 더하기와 곱하기가 정의되는 집합. <실수>와 동일한 방식으로 더하기와 곱하기가 정의되는 <집합>. ## 정의 A field is a $F$ together with two [binary operations](https://wiki.g15e.com/pages/Binary%20operation.txt) on $F$ called *addition* and *muliplication*. A binary operation on $F$ is a mapping $F \times F \rightarrow F$. - The addition of $a$ and $b$ in $F$ is called the sum of $a$ and $b$, and is denoted as $a + b$. - The multiplication of $a$ and $b$ in $F$ is called the product of $a$ and $b$, and is demoted as $a \times b$. The following properties are required to be satisfied: - of addition and multiplication: $a + (b + c) = (a + b) + c$, and $a \times (b \times c) = (a \times b) \times c$. - of addition and multiplication: $a + b = b + a$, and $a \times b = b \times a$ - of multiplication over addition: $a \times (b + c) = (a \times b) + (a \times c)$ - Additive identity and multiplicative identity: there exist two distinct elements 0 and 1 in $F$ such that $a + 0 = a$ and $a \times 1 = a$. - Additive inverse: for every $a$ in $F$, there exists an element in $F$, denoted by $-a$, called the additive inverse of $a$, such that $a - a = 0$ - Multiplicative inverse: for every $a \neq 0$ in $F$, there exists an element in $F$, denoted by $a^{-1}$ or $1/a$, called the multiplicative inverse of $a$, such that $a \cdot a^{-1} = 1$.