Reply To: Seven is the number of perfection
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you point out, that is governed by the odd or even condition of the denominator…
not really the odd/even condition, but whether there is a common factor between 10 and the divisor (when the fraction itself is in its ‘simplest’ form…)
fascinating that the 1/7 fraction repeats on a 6 digit pattern… is the denominator -1 a constant pattern?
It is, when the divisor is a prime number!
It could have been a 1 digit pattern, a 2 digit pattern, a 3 digit pattern, or a 6 digit pattern, because those are the divisors of 6.
Let me try to clarify this, without going into abstract algebra and group theory too much.
The following applies to all primes that are not 2 or 5 (because those are divisors of 10…), but we will concentrate on 7, for simplicity.
Now, 1/ 7 is something and 2/7 is something else and so on…but then something amazing happens: 7/7 = 1.00000000…
8/7 = 1 + 1/7
So, considering only what happens behind the decimal point, in the case of dividing by 7, the only interesting cases are:
1/7, 2/7, 3/7, 4/,7, 5/7, 6/7. And if you care to count them, that are 7-1 = 6 cases!
Let me now remind you of doing long division (I hope that is what you call them):
It might look something like:
7/1.00000…\ 0.142
7
30
28
20
etcetera…
If you know anything about modulo, you would recognize that 20 is congruent to -1 modulo 7 (3*7 – 1) and that means that 20² is congruent to 1 modulo 7
While writing this, I realize, that all this is going to be more confusing than enlightening, but let’s try a bit more…
Suppose, that 100000/7 would yield a rest of 1 (there would exist a natural number n, so that 100000 = n*7 + 1), then in the above long division, there would be a 1 at the fifth decimal, meaning the digits would start to repeat themselves…
but, they obviously repeat themselves after 6 digits (
‘So, considering only what happens behind the decimal point, in the case of dividing by 7, the only interesting cases are:
1/7, 2/7, 3/7, 4/,7, 5/7, 6/7. And if you care to count them, that are 7-1 = 6 cases!’
), so if they repeat after 5 and 6 digits, they must repeat after 1 digit. Try all cases and you will hopefully get a feel for why the
number of digits, when 1/p, with p being a prime, not equal to 2 or 5, repeats after a number of digits, that is a divisor of p-1.
Don’t worry if you don’t, I had a lot more mathematical knowledge when my math professor first showed this to me and I thought he was doing magic 😉