Do you know what one-seventh is as a decimal? It’s 0.142857 recurring, 142857 going on forever. If you calculated the other sevenths as fractions, you would find that they are rotations of those six digits, 285714, 428571, 571428, 714285 and 857142, each of them extending into infinity. The cyclic number might not be very exciting, but if you split each of those six sequences into two and add the two halves together, what do you get? 142+857 = 999, 285+714 = 999, 571+428 =999, 714+285=999, 857+142=999. I thought this was interesting.
I thought it even more interesting to discover that there were other numbers which had much longer sequences of recurring digits that still produced the 9s. One-seventeenth as a decimal is 0.0588235294117647, sixteen digits that when split into two groups of eight produce the sum 05882352+94117647 the answer to which is 99999999. “That’s neat,” I thought, “that’s really neat.”
Being a neophyte at mathematics, I was pleased to discover that there was a theorem, Midy’s theorem, that explains the recurrences and the patterns of numbers. (M.E. Midy was a French mathematician who published a 21 page pamphlet with the catchy title De Quelques Propriétés des Nombres et des Fractions Décimales Périodique in the city of Nantes in 1836). It’s probably stuff that’s now done in third year at secondary school, but to someone who took CSE maths in 1976 it was new and impressive.
Struggling to explain my shaky understanding of Midy’s theorem to someone, and pointing out how the sevenths produced the 9s, I received the response, “yes, but what use is it?”
It seemed an odd question, is mathematics so much the preserve of utilitarians that mathematical ideas must all have practical utility? Would the question, “what use is it?” be asked of other things? Would someone pick up a copy of Jane Austen’s Emma and say, “yes, but what use is it?” Would someone look at Van Gogh’s Sunflowers and say, “yes, but what use is it?” Would someone listen to Elgar’s Nimrod and say “yes, but what use is it?” Why should some things be worthwhile for their own sake while others are judged purely on the basis of their usefulness?
Beauty is always worthwhile for its own sake, it needs neither explanation nor defence. Beauty makes life different, it enhances, it transforms, it enriches, it pleases. Shouldn’t beauty be just as possible in sequences of numbers as in sequences of words on a page, or sequences of brush strokes on a canvas, or sequences of notes played by a musician? Sevenths can be beautiful.