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EXPLORING RATIONALITY, PART 2
IN MY CONTINUING exploration of rationality, I offer the completed Decimal Exploration table of rational numbers 1/2 through 1/17 expressed in decimal form.
Sorry about 1/17. I cannot speak for the reader, but I believe the exercise did me some good.
Now that we have our exploration data collected, let’s seek patterns and see if they lead us anywhere.
One pattern that jumps out is with powers of 2 on our list. Each terminates with a digit length corresponding to its power of 2. That is, 8 = 23, and 1/8 = .125, terminating with three decimal places; 16 is 24, and 1/16 = .0625 terminating in four. We might guess, then, that 1/32, being related to 25, would be a terminating decimal of five places. And indeed, it calculates out to .03125.
A similar argument pertains to 5 and its powers. No surprise, because our number system is base 10, 10 being 2 x 5.
The concept of decimals traces back to India, perhaps as early as the 5th century B.C. In the 9th century A.D., the Arabic mathematician Al-Khwārizmi favored decimal notation in his seminal work. The book’s name, partly al-jabr, gave us the word “algebra.” Al-Khwārizm lives on in our word “algorithm.”
Back to our table: Primes other than 2 and 5 all seem to repeat. And is it just coincidence that 1/7 repeats six digits and 1/17 repeats 16 of them?
Actually, 1/7 and 1/17 have special properties (shared by 1/13 too). Consider what happens when the digits of their decimal representations are split in two and added:
With 1/7’s repeating block 142857, 142 + 857 = 999.
With 1/17’s block 0588235294117647, 05882352 + 94117647 = 99,999,999.
Is this magic or what? No, it’s actually Midy’s Theorem. M.E. Midy was a 19th-century French mathematician who wrote a short (21-page) treatise in the 1830s titled De Quelques Propriétés des Nombres et des Fractions Décimales Périodique, On Some Properties of Numbers and Periodic Decimal Fractions. Midy’s work lay pretty much dormant until 2003 when a Yale undergraduate published an extension of his results. Since then, Midy’s Theorem has inspired modern number theorists, many of whom do fundamental work in cryptography and computer science.
For more on this, check out The Secret Theorem of M.E. Midy = Casting in Nines,” at “a mispelt bog” of John Kemeny. As the title suggests, the secret is related to the arithmetic process of “casting out nines.” Kemeny concludes his essay with a quotation from Mahatma Gandhi: “If only one person knows the truth, it is still the truth.”
Thanks for taking part in this little exploration of rationality. ds
© Dennis Simanaitis, SimanaitisSays.com, 2017
Simanaitis Says 
I have observed and deducted a few other relationships of interest.
Multiples such as 3×7, 5×7, etc. have the same number of recurring digits as 7.
Eleven has recurring block length of 2 for 22, 33, 44, 55, 66. For 77 though, the 7 repeating block length applies. 121 (11×11) blows up and I can’t find a repeating block length.
I would imagine that multiples of 13 and 17 would exhibit similar results.
I haven’t examined the non-recurring block for the above, but I’m sure that there would be a pattern too.
3 is another interesting relationship. Looking at 3^n the following pattern falls out:
– non-recurring block length is zero (I tested up to n=4)
– recurring block length is:
nrbl = 2^n – (2n-1), tested up to n=4
I wanted to test up to n=5, but Excel doesn’t carry enough digits to show the predicted 23 digit long recurring block, and I didn’t desire to do the longhand solution.
I could spend the rest of the weekend on this fun, but as it is my anniversary tomorrow the joy gained by success would be drowned out by the inevitable pain.
Oh yes, and the 6 digit recurring block for 13 when split and added together sums to 999 (076+923). Interesting that the repeating block for 13 is 6 long and not 12 as the pattern for 7 and 17 might suggest.
I imagine that primes like 19, 23, etc would exhibit similar properties, but block lengths get too large to test the theory.
Gold stars to you (and to the Mrs.).