Field definition math: A set S is called a field if the operations of addition ‘+ ’ and multiplication ‘. ’ on S satisfy
the following properties are written in tabular form:
| Addition |
| Closure for any set [katex] S\: a,b\in S \newline a+b \in S [/katex] Commutativity for any [katex]\: a,b\in S \newline a+b = b+a[/katex] Associativity for any [katex]\: a,b,c\in S \newline \left ( a+b \right )+c = a+\left ( b+c \right )[/katex] Existence of Identity for any [katex]a\in S\: \exists \: 0\in S[/katex] such that [katex]a+0=0+a=a[/katex] Existence of Inverses for any [katex]a\in S\:\: \: \exists \: -a\in S[/katex] such that [katex]a+\left ( -a \right )=\left ( -a \right )a=0[/katex] Distributivity for any [katex]a,b,c\in S \newline a\left ( b+c \right )=ab+ac \newline or\: \: \: \left ( b+c \right )a=ba+ca[/katex] |
| Multiplication |
| closure for any set [katex] S\: a,b\in S \newline a\times b \in S [/katex] Commutativity for any [katex]\: a,b\in S \newline a\times b = b\times a[/katex] Associativity for any [katex]\: a,b,c\in S \newline \left ( a+b \right )+c = a+\left ( b+c \right )[/katex] Existence of Identity for any [katex]a\in S\: \exists \: 1\in S[/katex] such that [katex]a.1=1.a=a[/katex] Existence of Inverses for any [katex]a\in S\:\: \:;a\neq 0\: \: \exists \:\: \: \frac{1}{a}\in S[/katex] such that [katex]a\left ( \frac{1}{a} \right )= \left ( \frac{1}{a} \right )a= 1[/katex] Distributivity for any [katex]a,b,c\in S \newline a\left ( b+c \right )=ab+ac \newline or\: \: \: \left ( b+c \right )a=ba+ca[/katex] |
All the above mentioned properties hold for [katex]\Re \: \: \:\: \mathbb{C}\: \: \: \: \mathbb{Q}[/katex].
Hence [katex]\Re \: \: \:\: \mathbb{C}\: \: \: \: \mathbb{Q}[/katex] are a field in math