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 ${\displaystyle 5{;}_3{2}_{12}4}$, 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 ${\displaystyle \frac{1}{3}}$.  This should not be a surprise---4 inches is a ${\displaystyle \frac{1}{3}}$ of a foot.  Add this to the 2 feet already there, to get ${\displaystyle 2\frac{1}{3}}$ feet, which can also be written ${\displaystyle \frac{7}{3}}$ feet.  Now move this ${\displaystyle \frac{7}{3}}$ left across the border again.  This time, we need to divide by 3, giving ${\displaystyle \frac{7}{9}}$.  Adding to what is already there, we get ${\displaystyle 5\frac{7}{9}}$.  Finally, then, we have no inches, no feet, and ${\displaystyle 5\frac{7}{9}}$ written in the yards rectangle.  So 5 yd., 2 ft., 4 in. is the same length as ${\displaystyle 5\frac{7}{9}}$ 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

${\displaystyle 15+2+\frac{1}{3} = 17\frac{1}{3},}$


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


Summarizing, then,


${\displaystyle 5_3{2}_{12}4 = 5_3{2}_{12}4;= (((5\times 3) + 2)\times 12) + 4}$
${\displaystyle = 208,}$

${\displaystyle 5_{3}2{;}_{12}4 = (5\times 3) + 2 + (12\backslash 4)}$
${\displaystyle = 17\frac{1}{3},}$

${\displaystyle 5{;}_3{2}_{12}4 = 5 + (3\backslash (2 + (12\backslash 4))) = 5 + \frac{2 + \frac{4}{12}}{3}}$
${\displaystyle = 5\frac{7}{9}.}$

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




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