Saturday, January 23, 2010

Wiggles general...

Consider the wiggle expression `2_{4}3_{5}4;_{6}5_{7}6`.  One way to calculate its value is to telescope from left and right towards the face value superdigit.  Telescoping from the left :

`2 xx 4  =  8,`

`8 + 3  = 11,`

`11 xx 5  =  55,`

`55 + 4;  =  59;`

Telescoping from the right :

`6 -: 7  =  \frac{6}{7},`

` \frac{6}{7} + 5  =  5 \frac{6}{7}  =  \frac{41}{7},`

` \frac{41}{7} -: 6 =  \frac{41}{42}.`

Combining these, we get

`2_{4}3_{5}4;_{6}5_{7}6  =  59 \frac{41}{42}.`

We shall eventually see a more efficient process for the fractional part of such problems than telescoping from the right, but not just yet.

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