• 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

**(3T ^{3} + 5T + T^{-1}) + (4T^{2} - 7T + 1) = 3T^{3} + 4T^{2} -2T + 1 + T^{-1} **

**(3T ^{3} + 5T + T^{-1}) * (4T^{2} - 7T + 1) = 12T^{5} - 21T^{4} + 23T^{3} - 35T^{2} + 9T - 7 + T^{-1} **

**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} **

**1 / (1-T ^{-k}) = 1 + T^{-k} + T^{-2k} + T^{-3k} + ..... + T^{-nk} + ...... **

**T / (T - 1) = 1 + T ^{-1} + T^{-2} + T^{-3} + ..... + T^{-n} + ...... **

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