On Cyclic Numbers and an Extension of Midy’s Theorem
نویسندگان
چکیده
In this note we consider fractions of the form 1 m and their floating-point representation in various arithmetic bases. For instance, what is 1 7 in base 2005? And, what about 1 4 ? We give a simple algorithm to answer these questions. In addition, we discuss an extension of Midy's theorem whose proof relies on elementary modular arithmetic. 1. Cyclic numbers and change of base Let us start with the simple and commonly used example p = 7. The number 1 7 = 0.142857 has a couple of fascinating properties that can be used to delight friends, even if they are familiarized with the mysteries of math. With the period 142857 we can associate the " key " 132645 which in this particular case represents the order of the digits 1 4 2 8 5 7 ↑ ↑ ↑ ↑ ↑ ↑ 1 3 2 6 4 5 so it indicates that 1 is the first digit, 4 is the third, 2 is the second, and so on.
منابع مشابه
A Graphical Analysis of Midy’s Theorem
Consider the fraction 1 7 . The decimal expansion of this fraction is .142857. At first glance, this may seem to be an ordinary decimal, but on closer examination, some interesting characteristics emerge. First, note that there is an even number of repeating digits. Then split the group of repeating digits into two “blocks”: 142 and 857. Then if we add these two blocks together we get 999. This...
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