Quasi-Rationals

``For every problem, there is one solution which is simple, neat and wrong."

H.L. Mencken (1880-1956)

Having established a visual character for irrationals and their expansions, it is interesting to note the existence of ``quasi-rational" numbers. These are certain well-known irrational numbers which generate images appearing suspiciously rational. The sequences pictured in Figures 9 and 10 are $\{i\pi\}_{i=1}^{1600}$ mod 2 and $\{i e\}_{i=1}^{1600}$ mod 2, respectively. One way of thinking about these sequences is as binary expansions of the numbers

$$\sum_{n=1}^{\infty} \frac{ [m\alpha] \mbox{ mod 2}}{2^i}$$

where $\alpha$ is, respectively, $\pi$ and e.

  

Figure 9: Integer part of $\{i\pi\}_{i=1}^{1600}$ mod 2; note the slight irregularities in the pseudo-periodic pattern.

The resulting images are very regular. And yet these are transcendental numbers; having observed this phenomenon, we were subsequently able to prove this rigourously from the study of

$$\sum_{n=1}^{\infty} \frac{ [m\alpha] }{2^i}$$

which is transcendental for all irrational $\alpha$. This follows from the remarkable continued fraction expansion of Böhmer [2]

$$\sum_{n=1}^{\infty} [m\alpha] z^n = \sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{(1-z^{q_n})(1-z^{q_{n+1}})}$$

Here (q_n) is the sequence of denominators in the simple continued fraction expansion of $\alpha$.

  

Figure 10: Integer part of $\{i e\}_{i=1}^{1600}$ mod 2; note the slight irregularities in the pseudo-periodic pattern.

Careful examination of Figures 9 and 10 show that they are only pseudo-periodic; slight irregularities appear in the pattern. This rational-like behaviour follows from the very good rational approximations provided by this expansion. Or put another way, there are very large terms in the continued fraction expansion. For example, the expansion of

$$\sum_{n=1}^{\infty} \frac{ [m\pi] mod 2}{2^i}$$

is

$$[0, 1, 2, 42, 638816050508714029100700827905 , 1, 126, \ldots]$$

with a similar phenomenon for e.

This behaviour makes it clear that there is subtlety in the nature of these numbers. Indeed, many related phenomena exist whose proofs are not yet in hand. For example, a similar result is not yet available for

$$\sum_{n=1}^{\infty} \frac{ [m\pi] mod 2}{3^i}$$

A proof for these graphic results might well offer further refinements to their representations, leading to yet another critical graphic characterization .