• Operations on infinite expansions

If we want to add or to multiply two formal numbers, one of them or both being represented by an infinite expansion, we will naturally operate on the truncated expansions to obtain an estimation for the result of this operation. Let us illustrate this with the two following formal numbers over the base field ** Q ** :

**X _{1} = 10.111111.....[1].... and X_{2} = 202.2121....[21]..... **

**X _{1} + X_{2} = 212.3232....[32]..... **

**X _{1} * X_{2} = 2044.588(11)(11)....... **

**X _{1} * X_{1} = 102.23456789(10)(11)....... **

• Fields of formal numbers

It is clear, as addition is defined, that every formal number has an opposite, namely the expansion obtained by taking the opposite of each coefficient. Now we will see that every non-zero formal number has an inverse. If ** X = a _{n}T^{n}**, we have already observed that its inverse will be

**X = a _{n}T^{n} ( 1 + a_{n}^{-1} a_{n-1}T^{-1}
+ a_{n}^{-1} a_{n-2}T^{-2} +......) = a_{n}T^{n} ( 1 - X_{1})
**

**1 / ( 1 - X _{1}) = 1 + X_{1} + X_{1}^{2}
+ ....... + X_{1}^{k} +...... **

**X ^{-1} = a_{n}^{-1}T^{-n} ( 1 + X_{1} + X_{1}^{2}
+ ....... + X_{1}^{k} +...... )
**

**(10.111111.....[1]....) ^{-1} = 0.10(-1)(-1)0110(-1)(-1)011.......[0(-1)(-1)011]...
**

Consequently, in the set of formal numbers we have the four operations : addition, multiplication, substraction and division. Thus this set is a field. The field of formal numbers over the base field