Irrational numbers

Since the rational numbers can be listed by the Farey-tree scheme, we say they are countably many, though there are in fact infinitely many of them. Also, there are infinitely many irrational numbers, but we have no knowledge of their full membership, so that we say the irrational numbers are uncountably many. In other words, we can at least tell the rational numbers, but we can't even tell what all irrational numbers are like. Some familiar examples of irrational are; is ratio of the circumference to diameter of a circle, is the diagonal of a unit square, e = lim_{n ->} (1+1/n)ⁿ ( 2.718) is the base of natural logarithms and appears in the exponential function key e^x of a pocket calculator, and the golden ratio = is what we have studied in Lesson 4.

Aside from such a technical subtlety, our main quest here is the decimal expression of irrational numbers. Although a pocket calculator shows = 3.141592654, we know that it is all the digits that it can display. In fact, the decimal expression of is never ending, and we present here the first 100 decimal digits as follows:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679…

Evidently, the decimal digits do not repeat themselves, regardless of how many decimal digits we write down explicitly for . And, this is also true for the decimal expression of the Golden ratio:

= 1.61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374…,

as well as any irrational number. Hence, the period of irrational numbers is infinitely long, and this is denoted by period . In any event, as period becomes long enough, it appears that the decimal digits have an irregular pattern, as exemplified in figure 8 by the first 21 decimal digits of It is indeed difficult to tell figure 8 apart from a random sequence of 21 numbers picked out of (0, 1,..., 9). But, here is a caveat. We are strictly speaking of the apparently random pattern of decimal digits of and, of course, its value is completely wrong if any portion of decimal digits is replaced by a random sequence.

Figure 8. First 21 decimal digits of

To sum up, all numbers have a period n, where n is the number of repeating decimal digits. Table 1 lists the rational fractions with n= 0, 1, ..., 6, and all rational numbers have a definite period. On the other hand, the period is infinitely long, that is, n becomes for irrational numbers.

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