LessonNumbers! Numbers are everywhere. Some are about us, such as,
the 4-digit birth year and 9-digit social security number. But, there are other numbers that nobody seems to know the exact value. Take, for instance, the value of Rational numbers The rational numbers are those we can express in fraction form I/J of two integer numbers I and J. For instance, I = 3 and J = 4, we have the fraction 3/4, which is the first row entry in table 1. Its decimal expression is 0.75, though zeros after the digit "5" are not usually shown. Next, the second-row fraction 1/3 has one digit repeating, the third-row fraction 7/11 has two repeating digits, and so on. As you may check with a pocket calculator, the number of repeating decimal digits increase up to 6 in table 1, and it is called the period of decimal expressions. We must however point out that many rational fractions can have the same period. For instance, (1/3, 2/3, 1/6, 1/9) have period 1 (1/11, 2/11, 7/11, 1/22, 1/33) have period 2, (1/27, 5/27, 1/37, 1/54, 1/74) have period 3, etc. These examples suggest that the period of I/J is determined not by numerator I but by the denominator J. It is therefore simple to examine the unit fraction I/J for period.
Table 1. Rational fractions
Beyond table 1, for example, the fractions 230/239 and 9/73 have the periods 7 and 8, respectively, and we can find a rational fraction of any period. This is possible because, in principle, we know all the rational numbers to search for a particular period of interest. (Project a - Farey tree for the rational fractions in (0, 1)). |
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