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We are often told to "put on a brave face” or to be strong. To do this, we often have to hide, or at least minimize, whatever fears, flaws, and vulnerabilities we possess. However, such an emphasis on strength is misguided. What truly takes courage is to show our imperfections, not to show our strengths, because it is only when we are able to show vulnerability - or the capacity to be hurt - that we are genuinely able to connect with other people.
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2007/05/28

並非巧合-偶然中的必然 (補完-MSN分享空間)

本文原本發表於MSN分享空間, 2007/05/28 19:58.

This post comes from previous post: 巧合.

Notation:
(1) a value of M-ary system is denoted as V_{M} and a value of decimal system is denoted as V_{10} or V for convenience.
(2) a value of M-ary system with D digits is written as
V_{M} = [v(D-1) v(D-2) ... v(1) v(0)]_{M} where 0 <= v(i) <= (M-1)
(3) V_{M} in decimal system is
V = (V_{M})_{10} = v(D-1)*M^(D-1) + v(D-2)*M^(D-2) + ... + v(1)*M + v(0)
(4) rev(V_{M}) means reverse order of V_{M}:
rev(V_{M}) = [v(0) v(1) ... v(D-2) v(D-1)]_{M}
(5) subtraction for two M-ary system values is analogous to decimal system.

Proposition:
For positive integer M > 1, any (2M)-ary system has
V_{M} = [(M-1) (M-2) ... 1]_{M} and U_{M} = [(M-1) (M-2) ... 1 0]_{M}.

Then we have (the subtraction is under (2M)-ary system)
   V_{M} - rev(V_{M}) = [(2M-2) (2M-4) ... 4 1 (2M-1) (2M-3) ... 3 2]_{M}
and
   U_{M} - rev(U_{M}) = [(2M-1) (2M-3) ... 3 0 (2M-2) (2M-4) ... 2 1]_{M}.

Exmaples:
(1) quaternary:
[3 2 1]_{4} - [1 2 3]_{4} = [1 3 2]_{4}
[3 2 1 0]_{4} - [0 1 2 3]_{4} = [3 0 2 1]_{4}
(2) senary:
[5 4 3 2 1]_{6} - [1 2 3 4 5]_{6} = [4 1 5 3 2]_{6}
[5 4 3 2 1 0]_{6} - [0 1 2 3 4 5]_{6} = [5 3 0 4 2 1]_{6}
(3) decimal:
[9 8 7 6 5 4 3 2 1]_{10} - [1 2 3 4 5 6 7 8 9]_{10} = [8 6 4 1 9 7 5 3 2]_{10}
[9 8 7 6 5 4 3 2 1 0]_{10} - [1 2 3 4 5 6 7 8 9 0]_{10} = [9 7 5 3 0 8 6 4 2 1]_{10}

By observing the examples, it's easy to figure out the rules.
The details of proof is not given.

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