Telescoping from the right, telescoping on the right, and full telescoping

Let's evaluate five weeks, two days, ten and three-quarter hours, in weeks and fractions of a week.

`\qquad\qquad\qquad 5;_{7}2_{24}10_{4}3.`

So far, we have followed used telescoping from the right, whereby the leftmost nonzero superdigit is divided by the interbase to its left, and then added to the superdigit to the left, this process being repeated until we are adding to the face value column.  Then


`\qquad\qquad\qquad 5;_{7}2_{24}10_{4}3  =  5;_{7}2_{24}(10\frac{3}{4}),`
`\qquad\qquad\qquad\qquad  =  5;_{7}2_{24}(\frac{43}{4}),`
`\qquad\qquad\qquad\qquad  =  5;_{7}(2\frac{43}{96}),`
`\qquad\qquad\qquad\qquad  =  5;_{7}(\frac{235}{96}),`
`\qquad\qquad\qquad\qquad  =  5\frac{235}{672}.`

Many of the practical uses for wiggle notation employ only integer or natural number interbases.  In these cases, the whole number part of the answer consists of the face value superdigit and the value of the expression to its left, while everything to the right produces the fractional part.  In such cases, instead of telescoping the fractional part from the left term by term as above, we can first telescope right...

`\qquad\qquad\qquad 5;_{7}2_{24}10_{4}3  =  5;_{7}0_{24}(2 xx 24  +  10)_{4}3,`
`\qquad\qquad\qquad\qquad  =  5;_{7}0_{24}58_{4}3,`
`\qquad\qquad\qquad\qquad  =  5;_{7}0_{24}0_{4}(58 xx 4  +  3),`
`\qquad\qquad\qquad\qquad  =  5;_{7}0_{24}0_{4}235,`

...and then compact left...

`\qquad\qquad\qquad\qquad  =  5\frac{235}{7 xx 24 xx 4}  =  5\frac{235}{672}.`

It can be written and remembered more succinctly :

`\qquad\qquad\qquad 5;_{7}2_{24}10_{4}3  =  5\frac{2_{24}10_{4}3}{7 xx 24 xx 4}  =  5\frac{235}{672}.`

It can be made more or less self-evident by the following argument :


`\qquad\qquad\qquad 5;_{7}2_{24}10_{4}3  =  5 + 0;_{7}2_{24}10_{4}3,`
`\qquad\qquad\qquad\qquad =  5 + 0;_{7}2_{24}10_{4}3,`
`\qquad\qquad\qquad\qquad=  5 + \frac{0;_{7}2_{24}10_{4}3}{1;_{7}0_{24}0_{4}0},` 
`\qquad\qquad\qquad\qquad =  5 + \frac{0_{7}2_{24}10_{4}3;}{1_{7}0_{24}0_{4}0;},`

`\qquad\qquad\qquad\qquad =  5 + \frac{2_{24}10_{4}3}{7 xx 24 xx 4},`
`\qquad\qquad\qquad\qquad =  5\frac{235}{672}.`

The combined process of telescoping the right part rightwards and then compacting it leftwards can be called telescoping on the right.  The arithmetic is perhaps a little simpler.

A slight variant of the process can be used for converting a wiggle expression to a pure rational form. In that case, one telescopes right not just the fractional part, but the whole wiggle expression, from its beginning (i.e. telescopes right, ignoring the semicolon), and then compacts left by dividing by the product of all the interbases to the right of the face value superdigit.  This can be called full telescoping.  The simplest case, and the most useful, is for converting a mixed fraction into a pure fraction :

`\qquad\qquad\qquad 3\frac{1}{7}  =  3;_{7}1  =  \frac{3_{7}1}{7}  =  \frac{22}{7}.

But the process also works for longer expressions.  e.g. to find the number of days, say, in five weeks, two days, ten and three-quarter hours, writing the result as a pure fraction,

`\qquad\qquad\qquad 5_{7}2;_{24}10_{4}3  =  \frac{5_{7}2_{24}10_{4}3}{24 xx 4},`
`\qquad\qquad\qquad  =  \frac{3559}{96}.`