Look closely, and you'll see that this fraction generates the first five Fibonacci numbers (1, 1, 2, 3, and 5) before blurring into other digits. Recall that, starting with 1 and 1, each successive Fibonacci number is the sum of the two previous Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.
Calculate 10000/9899. This time, you get 1.0102030508132134559046368… .
This fraction generates the first 10 Fibonacci numbers (using two digits per number). Going further, the fraction 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number).
Note that, in successive fractions, two 0s are appended to the numerator and a 9 to the beginning and end of the denominator.
Will the next fraction, 100000000/99989999, generate the first 20 Fibonacci numbers? Does the pattern continue forever? The answer appears to be yes.
James Smoak discovered this curious phenomenon, and he and Thomas J. Osler went on to prove that this class of fractions always produces decimal expansions containing terms of the Fibonacci sequence. They described it as "a magic trick from Fibonacci."
A little later, Marjorie Bicknell-Johnson found a formula, or "generalized mathematical magician," that identifies fractions whose decimal representations include successive values belonging to a variety of other sequences. She called them designer decimals.
Smoak (with O-Yeat Chan) then continued his adventures in the realm of designer decimals, as reported in the November 2006 College Mathematics Journal.
Consider, for example, the fraction 10000/9801. It has the decimal expansion 1.0203040506… , suggesting the existence of a new class of fractions with curious properties.
Smoak and Chan ask: Do all the integers from 1 to 99 occur in the sequence? Given that the decimal expansion must repeat, what is the length and nature of the repeating part?
The key, Smoak and Chan say, is to note that 9801 = 992. So 10000/9801 = (100/99)2 = (1.0101010101…)2.
Then, it's possible to show that the repeating part is 0203…97990001.
In general, fractions of the form [10n/(10n – 1)]k yield the sequence of integers in their decimal expansions.
It's amazing what can lie hidden in simple fractions!
Originally posted November 6, 2006
References:
Bicknell-Johnson, M. 2004. A generalized magic trick from Fibonacci: Designer decimals. College Mathematics Journal 35(March):125-126.
Chan, O-Y., and J. Smoak. 2006. More designer decimals: The integers and their geometric extensions. College Mathematics Journal 37(November):355-363.
Smoak, J., and T.J. Osler. 2003. A magic trick from Fibonacci. College Mathematics Journal 34(January):58-60.
1 comment:
Interesting fractions with interesting decimal expansions might be of interest to students of almost all ages and levels, and there are undoubtedly many articles accessible to high school, possibly middle school students.
See e.g. "Fun fractions? You've got to be kidding!", NCTM's Mathematics Teacher, November 1998 (v91), p572-575 (which starts with interesting aspects of 1/7, and lets readers create fractions with prescribed decimal expansions using just algebra, or calculus if they prefer, introducing the idea of generating functions). Another example is "Ascending and descending fractions", Mathematics Teacher, 2002 (v95) 539-542.
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