representation of real numbers
It is well-known that there are several methods to introduce the real numbers. We shall follow an inductive method which is instructive as well as elementary. Apart from that such treatment is modern, interesting and is obtained through two theorems and a lemma, which are relatively easy to understand. So that our aim will be to prove the above propositions. Our starting point is the following theorem.
Let be a sequence of integers and a sequence of real numbers defined by the equations
and for all
Next we multiply (4) by whence , but , so that
an inequality required by the theorem.
Now, by (3) and (4), and applying induction on , we can establish that
Now we define
thus from (4), (5) and by the hypothesis , we arrive to
because the fractional part . Then if we let grows beyond of any bound, will be so close to zero as we want and such a observation implies the representation (1).
We still need to prove the another inequality of the theorem, i.e. for infinitely many , but also the uniqueness of representation (1). To do that we need to make use of the identity
(It is legitimate to consider this identity as a lemma, as we need it to prove this theorem as well as the next one).
We shall prove later this identity. Let us prove the inequality by tertio excluso; thus suppose that there is a fixed integer such that for all . From (1) and (6) we get
and comparing this with (5) one realizes that , contradicting (4).
Finally we must prove the uniqueness of the representation (1). So that we suppose
so we see that is the integral part of , i.e. , but also from (2) , therefore . Next we shall again use tertio excluso. On this way let us suppose that for some the pair and are unequal. There is no loss of generality in assuming that is the smallest integer with this property (which is justified by a simple inductive argument), and that , so that . Thus we have
But then recalling that , we see tat . From this fact and (6), we can write
This is a contradiction with respect to the inequality found before, thus the proof of this theorem is complete. ∎
0.2 Some implications
First we remarked that Theorem 1 is a generalization of the standard decimal expansion for a real number . This may be seen by taking all the integers . Thus, if , (1) gives the decimal representation
Second, if , we must write its decimal representation on the form (7)and then changing all signs.
Third, any real could have an ambiguous decimal representation, e.g. , having an infinite successive sequence of ’s, which also involves a geometric series in in turns implying (at the limit) that also . For that reason, (7) represents that number with an infinite succession of ,s, that is, . The reason for this resides in that an infinite succession of ’s is ruled out by the condition of the theorem that for infinitely many , a condition that in the present example takes the form for infinitely many .
Now we prove (6).
For the integers sequence , where for every , we have
Theorem 1 also represents an irrational number whenever we add a couple of additional conditions. Thus we have the following important theorem.
We contradict the thesis by supposing is rational (, , coprime). By the last hypothesis in the preceding theorem, we can choose an integer sufficiently large in order to be a divisor of . Now we may use (1) replacing the LHS by our rational number assumption, next multiplying both side by the latter product, and rearranging terms we get (we do the partition sum )
By hypothesis, the LHS of (8) is obviously an integer. However, we have proved already that the last inequality of theorem 1 requires that for infinitely many , so that, from the RHS of (8) and the lemma we see that
a clear contradiction, proving the theorem. ∎
All we know there are different ways to prove the irrationality of . In particular, it results illustrative if we adapt our theorems and its hypotheses (all of which are true in this case) to this problem. From Taylor’s expansion
Let us use (1), by setting , , for all , and , . Thus,
- 1 I. Niven, IRRATIONAL NUMBERS, Ch. VII, pp. 83-88; also p. 157, The Mathematical Association of America, 2005.
|Title||representation of real numbers|
|Date of creation||2013-03-22 19:10:36|
|Last modified on||2013-03-22 19:10:36|
|Last modified by||perucho (2192)|