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• Operations on finite expansions

Let us illustrate these operations on two formal numbers with base field K = Q . Following the basic rules of algebraic computation, we can write

(3T3 + 5T + T-1) + (4T2 - 7T + 1) = 3T3 + 4T2 -2T + 1 + T-1

and

(3T3 + 5T + T-1) * (4T2 - 7T + 1) = 12T5 - 21T4 + 23T3 - 35T2 + 9T - 7 + T-1

If we use the simplified writing, we have to introduce brackets around the coefficients to avoid confusion when it is necessary. This becomes

3050.1 + 4(-7)1 = 34(-2)1.1

3050.1 * 4(-7)1 = (12)(-21)(23)(-35)9(-7).1

• Approximation by truncation

As in the case of the decimal expansion for a real number, a formal number represented by an infinite power series expansion will be estimated by the begining of this expansion. This estimation is getting more and more accurate by truncating the power series expansion further away. In other words, the tail of the expansion becomes more and more meaningless. In mathematical terms we say that a formal number is close to zero when its degree is a large negative integer. Let us give an application of this property. Considering two positive integers k and n, we can write the following classical identity

(1-T-nk) / (1-T-k) = 1 + T-k + T-2k + T-3k + ..... + T-(n-1)k

Since T-nk tends to zero when n tends to infinity, we obtain the infinite expansion

1 / (1-T-k) = 1 + T-k + T-2k + T-3k + ..... + T-nk + ......

For instance, taking k = 1, we can write

T / (T - 1) = 1 + T-1 + T-2 + T-3 + ..... + T-n + ......

or in simplified writing

10 / (10 - 1) = 1.1111.....1......

Observe that this formula is simply true looking at it in the classical context of real numbers ( i. e. 10 / 9 = 1.111111...1... ).

 

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