By changing ** T ** into ** T**^{7} in the above formula, we obtain
** 1 = 10000001*0.000000100000010000001....[0000001].... **

We recall that the blocks between square brackets are for ever repeated.
Consequently we can write
** 1011**^{-1} = 10111 * 10000001^{-1} = 10111 * 0.000000100000010000001....[0000001]....

which leads to
** 1011**^{-1} = 0.001011100101110010111....[0010111]....

This expansion is periodic in agreement with the property stated above since ** 1011**^{-1} is a rational number. Notice that, in analogy with the case of real numbers, the periodicity results from the fact that every formal integer over a finite field divides ** T**^{l} ( T^{m} -1) for some positive integers ** l** and ** m**.
Let us now look at some algebraic irrational numbers. In ** F(2)**, the equation ** X**^{2} = 1.1 has no solution, in other words the square root of ** 1.1 ** does not exist in ** F(2)**. Nevertheless the square root of ** 1.1 ** does exist in ** F(3)** and an approximation is ** 1.21121.... **.

Again in ** F(2)**, the equation ** X**^{3} = 1.1 has a unique solution, in other words the cubic root of ** 1.1 ** does exist and an approximation is ** 1.11100001.... **.
At last let us consider the following equation
If we work in ** F(2)**, this can be written
which leads to
** X = 0.1 + 0.1 * (0.1 + 0.1 * X**^{2})^{2} =0.1 + 0.1 * (0.01 + 0.01 * X^{4}) = 0.101 + 0.001 * X^{4}

Repeating the same arguments, we would obtain
** X = 0.1010001 + 0.0000001 * X**^{8}

By iteration we see that the above equation has a unique root ** X**_{2} in ** F(2)** and we have
** X**_{2} = 0.101000100000001000.....

where the sequence of the coefficients of ** X**_{2} is such that ** a**_{-i} is ** 1 ** if ** i ** is equal to ** 2**^{k} - 1 and ** 0 ** otherwise. It could be shown that the same equation has a unique solution in ** F(p) ** for all prime numbers ** p **. We give an approximation for the solutions ** X**_{3} and ** X**_{5} in ** F(3)** and ** F(5)** respectively.
** X**_{3} = 0.102020102000000... and ** X**_{5} = 0.1040200040302010......

The sequence of the coefficients for ** X**_{2} is very regular. Even though the sequences of the coefficients for ** X**_{3} and for ** X**_{5} do not appear to be as regular, it can be shown that those sequences are in a certain sense not completely at random. Indeed mathematicians have defined a special class of sequences taking only a finite number of values. These sequences, called automatic sequences, are in a way a generalization of ultimately periodic sequences. A sequence is automatic if and only if it is the sequence of the coefficients of an algebraic formal number over a finite field. Consequently it is possible to decide whether a given formal number is algebraic or transcendental. Let us consider in ** F(p)** the following number
** X = 0.0110101000101000101000100000101.... **

where the sequence of the coefficients of ** X ** is such that ** a**_{-i} is ** 1 ** if ** i ** is a prime number and ** 0 ** otherwise. Then it can be proved that this number ** X ** is transcendental in ** F(p)** for all prime numbers ** p**. It is highly probable that the real number represented by the same expansion, considered as its decimal expansion, is also transcendental but the proof of that is yet out of reach.