One of my more distinct recollections of math class in grade school 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 happened to come across an article by Francesco Calogero of the University of Rome "La Sapienza" in the Fall 2003 issue of 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) 0.12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455. . . .
Try the fraction 1000/998001, or (23 x 52)/(36 x 372).
In decimal form, it has the value (to 100 decimal places) 0.001002003004005006007008009010011012013014015016017018019020021022023024025026027028029030031032033034. . . .
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!
Originally posted Nov. 17, 2003
Calogero, F. 2003. Cool irrational numbers and their rather cool rational approximations. Mathematical Intelligencer 25(No. 4):72-76.