The body of the numbers
The set of numbers is what is called a body (sometimes the field designation is also used)
A body is a set with two operations \displaystyle +\text{ and }\cdot (called sum and product), defined on F so that they fulfill the following properties:
- Associative property for addition \displaystyle (a+b)+c=a+(b+c)\text{; }\forall a, b, c\in F
- Commutative property for addition \displaystyle a+b=b+a\text{; }\forall a, b\in F
- Neutral element of the sum
There is some element \displaystyle a+b=b+a\text{; }0\in F such that \displaystyle a+0=a \text{; }\forall a\in F
For all \displaystyle a\in F, exist some element \displaystyle b\in F such that \displaystyle a+b=0 (that b is the reverse with respect to the sum from a, which is denoted by −a and is often referred to as the opposite element of a) - Associative property for the product
\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)\text{; }\forall a, b, c\in F - Commutative property for the product
\displaystyle a\cdot b=b\cdot a\text{; }\forall a, b\in F - Neutral element of the product
There is some element \displaystyle 1\in F such that \displaystyle a\cdot 1=a\text{; }\forall a\in F
For all \displaystyle a\in F with \displaystyle a\not=0, exist some element \displaystyle b\in F such that \displaystyle a\cdot b=1 (that b is the reverse with respect to the product from a, which is denoted by \displaystyle a^{-1})
When the property of the existence of inverse with respect to the product fails, instead of body, we have what is called ring (which has its own properties, which we are not going to mention at this time, but due to the lack of an inverse, they differ a lot from the multiplication that is usually taught in primary school)
For example, natural, real, and complex numbers are not bodies. Another set that does not form a body are polynomials (with rational, real or complex coefficients), but it is formed by the so called rational functions, that is, the quotients of polynomials