Compound addition, carrying, and standard form

We have looked at converting mixed unit expressions into single-unit expressions, e.g.

`\qquad\qquad\qquad 5 "yd.", 2 "ft.", 4 "in."  =  17\frac{1}{3} "ft."`

It can be useful, and also instructive, to perform arithmetic directly with mixed expressions.  This is called compound arithmetic, and was once a substantial part of elementary arithmetic.

Addition is the most elementary of these arithmetical operations.  Suppose, then, that we wish to add 2 yards, 2 feet, 9 inches to 5 yards, 2 feet, 4 inches.

The simplest way to approach this is to add like terms, so that

`\qquad\qquad\qquad (2 "yd." + 2 "ft." + 9 "in.")  +  (5 "yd." + 2 "ft." + 4 "in.")`
`\qquad\qquad\qquad\qquad  =  (2 + 5) "yd."  +  (2 + 2) "ft."  +  (9 + 4) "in."`
`\qquad\qquad\qquad\qquad  =  7 "yd." + 4 "ft." + 13 "in."`

This result, while true, is unfinished.  It is customary to combine smaller units into whole bigger unit equivalents wherever possible, in a process equivalent to carrying in elementary arithmetic :

`\qquad\qquad\qquad 7 "yd." + 4 "ft." + 13 "in."  =  7 "yd." + 4 "ft." + 1"ft." + 1 "in." `
`\qquad\qquad\qquad\qquad =  7 "yd." + 5 "ft." + 1" in."`
`\qquad\qquad\qquad\qquad =  7 "yd." + 1 "yd." + 2 "ft." + 1 "in."`
`\qquad\qquad\qquad\qquad =  8 "yd." + 2 "ft." + 1 "in."`

After carrying has been completed, the result is in standard form.   

Carrying and standard form also apply to wiggle notation in an obvious way, at least when the interbases are all whole numbers greater than one and the total quantity is not negative.  Then, wiggle notation is in standard form when each superdigit is non-negative, each superdigit except (perhaps) the last is a whole number, and each interbase is greater than the superdigit to it's immediate right.

There are at least two reasons to prefer standard form.  The first is that making full use of larger units keeps the corresponding numbers of units smaller.

The second, more important, reason is that it makes for ease of comparison---when quantities are in standard form, we can tell when one quantity is larger than another, or if they are equal.  One compares the amounts of the largest unit, and if those are equal, one compares the amounts of the next largest unit, and so forth, until we knows.  The process is similar to determining the order of words in a dictionary---lexical ordering.

It can reasonably be argued that both an ordered alphabet (or ordered syllabary) and either a place value number system or (more likely) the use of mixed units in standard form are probable antecedents to the invention of lexically ordered dictionaries.