An introduction to wiggle notation.

This afternoon, my day-and-a-half-old boy was weighed again.  He is now 7 lb., 8 oz.  (Newborns usually lose weight for the first little bit.)  So how many ounces is he?  Following the setup on the preceding post, we can calculate :

`\qquad\qquad\qquad (7  xx  16)  +  8  =  120,`

i. e., today, he weighed in at 120 oz.

We are going to write the left hand side of the calculation in a new way :

`\qquad\qquad\qquad 7_{16}8 = 120.`

This is an example of wiggle notation.  The 7 and the 8, written on the line, are called superdigits when we have to be precise, or digits when there is no risk of confusion. The 16, written at the level of a subscript, is called an interbase when we have to be precise, or else a base when there is no risk of confusion.  In general, a superdigit can be any quantity, and so can an interbase.  The justification for these names must wait.

To see how wiggle notation works, we should use an example with more superdigits and interbases.  Consider, then, the quantity 5 yd., 2 ft., 4 in.  How many inches is this?  Since there are 3 feet in a yard, and  12 inches in a foot, there are 36 inches in a yard.  One way to calculate the total number of inches is to convert each unit directly to inches.  Using the number convention, we get :

`\qquad\qquad\qquad (5 xx 36)  +  (2 xx 12)  +  4  =  208.`

Another way is to convert the given yards into feet, add to the feet already given, and then convert that into inches and add the given inches:

`\qquad\qquad\qquad (((5 xx 3)  +  2)  xx  12)  +  4  =  208.`

You get 208 inches either way, of course.

The left hand side can be written in wiggle notation, with :

`\qquad\qquad\qquad 5_{3}2_{12}4  =  208.`

Wiggle notation consists of a series of superdigits, separated by interbases written at level of subscripts.  Both ends of a wiggle expression are superdigits, written at the regularly.  Between them, the numbers wiggle down and up, like a V, or a W, or ...;  there is always one fewer of the interbases than the superdigits.

In `5_{3}2_{12}4` the (super)digits are 5, 2, and 4.  The (inter)bases are 3 and 12.

Calculation proceeds left to right, multiplying by bases, and adding digits:

`\qquad\qquad\qquad 5 xx 3  =  15,`

`\qquad\qquad\qquad 15+ 2  =  17,`

`\qquad\qquad\qquad 17 xx 12  =  204,`

`\qquad\qquad\qquad 204 + 4  =  208.`