More boxed wiggle

Now, as a second exercise, let's using the counting board again to convert  5 yd., 2 ft., 4 in. completely into yards.  This time, the relevant wiggle notation is `5;_{3}2_{12}4`, with the semicolon now marking the leftmost superdigit.  What we do with the counting board will help us understand what to do with this wiggle notation.

First, move the 4 left across the border.  Since we are moving left, we divide by the 12 at the border.  The 4 becomes `\frac{1}{3}`.  This should not be a surprise---4 inches is a `\frac{1}{3}` of a foot.  Add this to the 2 feet already there, to get `2\frac{1}{3}` feet, which can also be written `\frac{7}{3}` feet.  Now move this `\frac{7}{3}` left across the border again.  This time, we need to divide by 3, giving `\frac{7}{9}`.  Adding to what is already there, we get `5\frac{7}{9}`.  Finally, then, we have no inches, no feet, and `5\frac{7}{9}` written in the yards rectangle.  So 5 yd., 2 ft., 4 in. is the same length as `5\frac{7}{9}` yards.

As a third exercise, we convert 5 yd., 2 ft., 4 in. completely into feet.  Starting as before, we move the 5 rightward, multiplying by 3, and we move the 4 leftward, dividing by 12.  Adding to the 2 already there, we get

`\qquad\qquad\qquad 15 + 2 + \frac{1}{3}  =  17\frac{1}{3},`


so that 5 yd., 2 ft., 4 in. is `17\frac{1}{3}` feet.


Summarizing, then,


`\qquad\qquad\qquad 5_{3}2_{12}4  =  5_{3}2_{12}4; =  (((5 xx 3)  +  2) xx 12)  +  4`
`\qquad\qquad\qquad\qquad   =  208,`

`\qquad\qquad\qquad 5_{3}2;_{12}4  =  (5 xx 3)  +  2  +  (12 \\ 4)`
`\qquad\qquad\qquad\qquad   =  17\frac{1}{3},`

`\qquad\qquad\qquad 5;_{3}2_{12}4  =  5  +  (3 \\ (2  +  (12 \\ 4)))  =  5  +  \frac{2  +  \frac{4}{12}}{3}`
`\qquad\qquad\qquad\qquad   =  5\frac{7}{9}.`

From these arithmetical calculations, we see that we have arranged things so that

`\qquad\qquad\qquad 5" yd." + 2" ft." + 4" in." `
`\qquad\qquad\qquad\qquad  =  5_{3}2_{12}4;" in."  =  5_{3}2;_{12}4" ft."  =  5;_{3}2_{12}4" yd."

The unit maker, ";", behaves something like a decimal point.  As we shall soon see, we can make this analogy much stronger.