Cool Rationals

It’s curious how some classroom words, activities, or incidents can stick in your mind for years. I can still recall certain grammar rules from lessons long past, for example. When one of these rules comes into play as I write, I can remember not only the teacher’s words but also the tone and manner in which the rule was first presented.

One of my more distinct recollections of math class involves the decimal representation of rational numbers and the discovery of wonderful patterns among those digits.

Consider the fraction 1/7. Expressed as a decimal, it has the form 0.142857142857. . . , where the digits 142857 are repeated, ad infinitum. The surprise to me as a child was learning that the fraction 2/7 has the same decimal digits but in a different order: 0.285714285714. . . . This is also true for 3/7 (0.4285714285714. . . ), 4/7 (0.571428571428. . . ), 5/7 (0.714285714285. . . ), and 6/7 (0.857142857142. . . ).

To my young mind, that was an amazing, inexplicable pattern—a glimpse into the mysteries of numbers. What made the thrill of discovery even stronger for me was how the digits emerged one by one as I laboriously performed the long division operations needed to get the answers. There were no calculators in those days, and it’s possible that using a calculator would have eliminated much of the suspense and surprise.

I was reminded of this scene from my distant past when I recently saw an article by Francesco Calogero of the University of Rome “La Sapienza” in the current issue the Mathematical Intelligencer.

In his report on “cool” irrational numbers and their rational approximations, Calogero starts off with the example of 10/81. Expressed in decimals, this fraction has the value 0.123456790, with these digits endlessly repeated in the same order. Only the digit 8 is missing from the sequence.

According to Calogero, that defect can be corrected by subtracting from 10/81 a number of order 10–9 so as to change the last two of the first nine decimal digits from 90 to 89. He comes up with the following expression:

10/81 – 10–9 (3340/3267).

In decimal form, it has the value (to 101 decimal places) 0.1234567891011121314151617181920


383940414243444546474849505152535455. . . .


Try the fraction 1000/998001, or (23 x 52)/(36 x 372).

In decimal form, it has the value (to 100 decimal places) 0.001002003004005006007008009010011012013014015


1032033034. . . .

In his article, Calogero goes on to provide an explanation for such “numerology” and offers several additional examples of numbers that display remarkable patterns when written out in decimal form.

Ah, sweet mystery of rational number!

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