
Re: 0.9999... = 1 that means mathematics ends in contradiction
Posted:
Mar 14, 2013 10:09 PM


On Mar 13, 1:25 pm, JT <jonas.thornv...@gmail.com> wrote: > On 13 mar, 19:47, fom <fomJ...@nyms.net> wrote: > Dec NyaNTern StandardTern > 1 =1 01 > 2 =2 02 > 3 =3 10 > 4 =11 3+1 11 > 5 =12 3+2 12 > 6 =13 3+3 20 > 7 =21 6+1 21 > 8 =22 6+2 22 > 9 =23 6+3 100 > 10 =31 9+1 101 > 11 =32 9+2 102 > 12 =33 9+3 110 > 13 =111 9+3+1 111 > 14 =112 9+3+2 112 > 15 =113 9+3+3 120 > 16 =121 9+6+1 121 > 17 =122 9+6+2 122 > 18 =123 9+6+3 200 > 19 =131 9+9+1 201 > 20 =132 9+9+2 202 > 21 =133 9+9+3 210
I don't post much here any more, but I wanted to post at least once here on Pi Day. And so, in honor of Pi Day, I consider, how would we write the number pi in the bijective numeration system NyaN?
Decimal is an interesting case, since the first zero occurs rather late in the expansion. In standard decimal we see that pi begins:
3.1415926535897932384626433832795028841971693993751058209...
In NyaN, this becomes (using JT's suggested X for ten:
3.1415926535897932384626433832794X2884197169399374XX581X9...
We notice that the string "510" becomes "4XX" in NyaN.
In ternary, which appear to be JT's preferred base, we have that pi in standard ternary is:
10.0102110122220102110021111102212222201112012121212001...
I forget how JT explained to write fractions less than 1/2 in NyaN (or, in general, less than 1/(n1)  it's because of this that NyaN is awkward to use with real numbers).
This is what I obtained, starting with two '_' symbols:
3.__31333122212331332313333332212222131111312121211331...
(Notice how the string 2110 becomes 1333, and the even longer 211110 becomes 133333.)
Binary is an especially tricky case. We notice that with the natural numbers in binary, only the _repunits_ are identical in both standard and NyaN. All other naturals become one digit shorter in NyaN compared to standard binary.
In fact, this rule appears to work converting binary to NyaN:
 Replace the rightmost 0 with 2.  Drop the leftmost 1.  Increase all digits in between by 1.
With irrational numbers like pi, we simply ignore the first statement above, since there is no rightmost digit.
And so we see that:
11.0010010000111111011010101000100010000101101000110000...
becomes:
11.___1121111222222122121212111211121111212212111221111...
Hmmm. I notice that in NyaN binary
._22222222...
is already 1! (And to think that the OP has a problem with .9999...=1 in decimal, compare this to ._2222...=1 in the binary form of NyaN.)
Anyway, Happy Pi Day, everyone!

