Order in Chaos: decimal expansions of rational numbers
Though seemingly simple in nature, decimal expansions of fractions reveal many beautiful
properties of numbers. These decimals come in two forms:
• terminating decimals, such as 0.5, 0.001, and 0.512,
• repeating decimals, such as 0.3, 0.083, and 0.142857.
A first natural question arises: which fractions have terminating decimals and which have
repeating decimals? Why?
A second, deeper question quickly presents itself: when does the decimal expansion of
repeat/terminate? For the repeating decimals, paths for further investigation abound:
a
b
1. Does the length of the repeating decimal depend on a or only on b (assuming the fraction
is reduced)?
2. When does the entire decimal repeat (as in 0.142857) versus only part of the decimal (as
in 0.083)?
3. When is the repeating decimal as long as possible, as determined by the denominator b?
4. Can you predict the length of the repeating decimal without finding the decimal expansion?
5. Are there restrictions on what the repeating decimal’s length can be?
6. In what ways do the answers to this question change if, instead of the decimal system,
we were to use the binary number system?
We will investigate a few of these issues and return to this surprisingly rich topic later,
hopefully in Taos this summer. The goal is to obtain a fresh, rich understanding of the
structure inherent in decimal expansions of rational numbers: as a result, these seemingly
random sequences of numbers will hopefully take on a new meaning!
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