Askthepizzaguy
means x[sup]2[/sup] - (x-1)[sup]2[/sup] + 1 = 2x
It was something I noticed about numbers when I was 6 or 7 years old, back when I thought learning was still fun.
School taught me otherwise. 
Nice.
Right. School can be about cramming many times which is understandable given the amount of stuff thats needed to be taught in limited time and plus number of students in class.
Another magic trick based on 1/7 – though it uses only integer arithmetic, not fractions, involves preparing a paper slip with 142857 glued into a loop. Your target picks a number, does some arithmetic, and – presto – sees his number after you apply scissors judiciously.
7 works because it divides 999999 (six 9’s), but no shorter 9-strig; 7 = 1+six
Similarly, 17 divides 9999999999999999 (sixteen 9’s), but no shorter 9-string; 17 = 1+sixteen
1/13 repeats only 6 times as it marches 2/13, 3/13 etc. 13 divides twelve 9’s but it also divides six 9’s.
Yes, in fact, if you kept doubling and moving 2 digits to the right you’d get
0.14
0.0028
0.000056
0.00000112
0.0000000224
...
These sum to 1/7. The spillover from 112 being 3 digits long is what explains getting 57 instead of 56, and so on.
Why do these sum to 1/7? Well, what do you get when you start with x and keep doubling and moving 2 digits to the right? You get k, where k = x + 2k/100; that is (applying basic algebra), you get 50/49 * x. In the specific case where x = 0.14 (i.e., 7/50), this comes out to 1/7.
No, they mean cyclical numbers in the sense that non-integer multiples have the same periodic string of digits after the decimal point, only shifted. And, yes, every cyclical number in this sense is a fraction whose lowest-terms denominator is prime (or 1).
Why is that? Well, first of all, the fractional digits of a number are periodic just in case its fractional component is a multiple of a string of the form 0.000100010001… [with matching period]. And such a string satisfies the equation 10[sup]period[/sup] * x = 1 + x, which can be solved to get x = 1/(10[sup]period[/sup] - 1); thus, a string has periodic fractional component just in case it is a rational number whose lowest terms denominator divides the difference between 1 and some positive power of 10 (i.e., just in case there is a positive power of 10 which is 1 modulo this denominator; i.e., just in case the powers of 10 form a periodic sequence modulo this denominator).
Next, writing out formally the definition of cyclicality, we see that such a fraction m/n in lowest terms is cyclical just in case for every integer f such that f * m/n is not an integer, we have that there is some natural number p such that f * m/n - 10[sup]p[/sup] * m/n is an integer. That is (since m and n are coprime), every integer which is not divisible by n is a power of 10 away from a multiple of n. This amounts to the same thing as saying that every nonzero remainder modulo n is achieved by some power of 10.
Finally, why does this mean n cannot be composite? Well, we’ve now established that the powers of 10 form a periodic sequence which goes through every nonzero remainder modulo n (this sequence must also never hit zero modulo n (unless n = 1), or else it would stop being periodic and just be zero from then on out, never returning to its starting value of 1 (unless 1 = 0 modulo n, as only happens when n = 1)). Now assume for sake of contradiction that n is composite; then there are a and b less than n such that a * b = n. But, modulo n, the left hand side is a product of nonzero values and thus a product of powers of 10 and thus itself a power of 10 and thus nonzero, but the right-hand side is zero. This is a contradiction, establishing that n cannot be composite.
In base 2, 1/7 (or 1/111 to be consistent) is 0.001001001… That happens because it is 1 less than the cube of the base. Similarly, 1/11 (i.e. one third) = 0.010101…
The weird decimal expansion that most impresses me is:
1/37 = 0.027027027…
1/27 = 0.037037037…
(From Charles Fleischer’s classic Moleeds routine that is chock full of this stuff. Start a little after 14:00 in.)
I haven’t watched the routine, but this can be explained by the fact that 27 * 37 = 999. Any other factorization of a string of 9s gives the same sort of thing, because (dare I say it?) 0.999999… = 1.
Learning still is fun. Don’t let school convince you differently.
Cool, so for 99999 you have 271 and 369 whose inverses are 0.0036900369… and 0.0027100271…, resp.
(That 271 is 2710+1 and 3710-1 appears to be a semifluke. The next several blocks of 9s don’t seem to have similar factors.)
There’s something evil about 9s I tellz ya.
Right, similarly, 1/9 = 0.1111…, and 1/11 = 0.09090909…, because 9*11 = 99.
It’s just because they’re 1 less than the number base we use. If we used base nine, then we’d notice similar eerie patterns about 8.
Well, semi. Whenever A * B = a string of 9s, with B - A = 10, you’ll have that (A * 10 + 1) * (B * 10 - 1) = 100AB + 10(B - A) - 1 = …99900 + 100 - 1 = …99999 is also a string of 9s. [And the analogue in any base, taking string of 9s to mean 1 less than a power of 10; everything I’ve said in this thread has been essentially base-independent]
Also similarly: 1/9 = 0.1111…, and 1/1 = 0.9999…, because 9 * 1 = 9.
You’re just begging for a hijack aren’t you? 
Or, for that matter, 1/3 = 0.333… and 1/3 = 0.333…
A better question is:
Are cyclical numbers always non-constructable-by-compass-and-straightedge polygons?
Edit: ignore this post.
1/7 is constructible.