Boxes of wiggle

Let's look again at the length we considered before, i.e. 5 yd., 2 ft., 4 in.

Do or imagine the following:

Take a whiteboard and an indelible marker.  Left to right, draw a strip of three rectangles, so the each rectangle abuts the next.  Write the largest unit, yards, above the first rectangle, feet above the second, and the smallest unit, inches, above the third.  Just below the strip, under the border between the yards rectangle and the feet rectangle, write the number of feet in a yard, i.e. 3.  Just under the border between the feet rectangle and the inches rectangle, write the number of inches in a foot, i.e. 12.

What we have just made is a counting board, specialized to help us solve problems using yards, feet, and inches.  Counting boards have a long history, going back through medieval times and well into antiquity.

Put away the indelible marker and take out a dry-erase marker.  We now get to use the counting board that we have made.

As a first exercise, we are going to convert 5 yd., 2 ft., 4 in. completely into inches, as we did before.  Write 5 in the yards rectangle.  Write 2 in the feet rectangle.  Write 4 in the inches rectangle.  What we have just written means 5 yd. + 2 ft. + 4 in.

The first step is to convert the yards into feet.  We need to move the 5 yards to the feet rectangle.  Move the 5 toward the feet rectangle.  As it crosses the border, the 5 is multiplied by the 3 that marks the borderline.  It arrives in the feet rectangle as 15.  No surprise there---5 yards is the same length as 15 feet.  There is already a 2 there.  The 15 and the 2 add to give 17.  So now we have nothing---or zero---in the yards rectangle, 17 in the feet rectangle, and still 4 in the inches rectangle.

The second step is to convert all these feet to inches.  Move the 17 across the border rightwards, multiplying by 12 as we go.  This gives 204.  Again, no surprise.  17 feet is the same length as 204 inches.  Add this to the 4 inches already there, giving 208 inches.  In the end, we get nothing in the yard rectangle, nothing in the feet rectangle, and 208 in the inches rectangle.

What we do with the counting board is exactly what happens when we calculate using the wiggle notation `5_{3}2_{12}4`, also known as `5_{3}2_{12}4;`.