• 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 :
X1 = 10.111111.....[1].... and X2 = 202.2121....[21].....
Concerning addition, it is clear that the coefficient of a term of degree m in the sum is obtained by adding the coefficients of the terms of degree m in both numbers. Thus we will simply have
X1 + X2 = 212.3232....[32].....
Concerning multiplication, it is not so simple. Nevertheless using the product on finite expansions we obtain the following rule. The coefficient of a term of degree m in the product is obtained by adding all the products of the coefficients of the terms of degree i in one number and of degree j in the other one such that i + j = m . With the above two numbers, we have the following estimation for their product
X1 * X2 = 2044.588(11)(11).......
Let us give a last example where the sequence of the coefficients is very regular
X1 * X1 = 102.23456789(10)(11).......
Here the sequence of the coefficients onward is just the sequence of the natural integers. It is interesting to notice that the formal number X12 = (T +1/ (T-1))2 is indeed a rational number. Nevertheless the sequence of its coefficients is not ultimately periodic.
• 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 = anTn, we have already observed that its inverse will be X = an-1T-n. If X is neither zero nor equal to anTn then we can write
X = anTn ( 1 + an-1 an-1T-1
+ an-1 an-2T-2 +......) = anTn ( 1 - X1)
where X1 is a formal number of negative degree. Since X1k tends to zero as k tends to infinity, we have the following equality
1 / ( 1 - X1) = 1 + X1 + X12
+ ....... + X1k +......
and therefore
X-1 = an-1T-n ( 1 + X1 + X12
+ ....... + X1k +...... )
For example we obtain
(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 K can be denoted by F(K) . This field is analogous to the field R of real numbers. It contains two important subsets. The first one is the set of formal integers, in other words polynomials in T with coefficients in K , usually denoted by K [T] and corresponding to Z . The second one is the set of formal rational numbers, that is to say quotients of formal integers, usually denoted by K (T) and corresponding to Q .