This time I resorted to pen and paper to find out if 1/17ths and 1/19ths have recurring sequences when described as decimals. I’m not sure I could have kept a long sequence in my head, and the calculator apps I have do not run to enough decimal places to spot the pattern, in fact it is much easier to ‘see the pattern coming’ working manually for reasons I stated in my last post.
1/17 = 0.058823529417647058823529417647……
with a 16 digit long sequence
1/19 = 0.052631578947368421052631578947368421
with an 18 digit long sequence
So far my hunch is right, though it would be good to have a deductive rather than inductive method of proof. Another pattern that seems to be emerging for these vulgar fractions with prime number denominators is that the recurring sequences are 1 less than the denominator. n/7ths have a 6 number sequence and n/13ths have two sequences of 6 numbers (6+6=12).
n/11ths (0.09090909…., 0.18181818…., etc), n/3rds, n/5ths (which have a simple relationship to the decimal system) fall outside this pattern. Though it could be argued that there 10 separate recurring number sequences.
Next steps would be to look at n/23rds, n/39ths, n/31sts. Working in different bases. My other hunch is that this is something akin to addition of oscillating waves, a result between the interaction of two base systems; that of the denominator and that of the system within which it is described.
From my time many years ago as a mathematic student writing a piece of software in Fortran to calculate the area under the graph of a particularly complex equation I discovered how difficult it is for binary to describe fractions that decimal does easily, for example
1/5 in base 2 = 0.00110011…….
The version of Fortran we were using had a fixed point rather than a floating point number system and initial coding led to wild results.
Another thing I have noticed (though this should be obvious but nine the less interesting) is the sequence remains intact in integer multiplication, or when added to themselves except the result is an integer. As a sequence they almost have a DNA like predilection for self replication.
Anyway this may seem all rather tedious to some of you on this day after a General Election but the conundrums of numbers will last on into the future much longer than present woes and help me keep some perspective on it all.
More to follow.