A second cut at negative quantities---complementary representations

Fashioning negative quantities in sign and magnitude form, as we did in the last post, requires negation, so called because it makes positive quantities negative.

This is probably not the best of names, because negation also makes negative quantities positive.  It also changes left to right, north to south, east to west, debit to credit, and vice versa in each case.  There is nothing inherently positive or negative about east and west, but they are clearly opposites, and negation turns the one into the other. 

So a better name might be something like turning, reversal, or flipping, but we are probably stuck with negation---the negative of east is west, and vice versa.

Here, however, we are dealing with quantities that do have a sign.

The negative in sign and magnitude form of my littlest's birthweight, with the magnitude expressed in mixed units, is :

`\qquad\qquad\qquad -(7" lb." + 12" oz.").`

Negating each of the terms separately, we get

`\qquad\qquad\qquad -(7" lb." + 12" oz.")  =  \bar{7}" lb."  +  \bar{12}" oz."`

This is what we would use in calculations.  It turns out, however, not to be the only form, or even the most convenient.

Complementary representations of negative quantities are particularly convenient for addition.  In a complementary representation, we allow only the amount of the largest unit to be negative, requiring all the other parts of the mixed quantity to follow the usual rules for standard form, including requiring them to be non-negative.  We get the complementary form of  `\bar{7}" lb."  +  \bar{12}" oz."` by borrowing (when already in debt...) :

`\qquad\qquad\qquad \bar{7}" lb."  +  \bar{12}" oz."  =  \bar{7}" lb."  -  1" lb."  +  16" oz."  +  \bar{12}" oz.,"`
`\qquad\qquad\qquad\qquad =  \bar{8}" lb."  +  4" oz."`

Now that we have the negative of my son's birthweight, in complementary form, we can use it to calculate his growth from birth at his other weighings.  Instead of subtracting his birthweight from his subsequent weights, we can add the negative of his birthweight in this form to these subsequent weights.

So my littlest's growth by Friday is

`\qquad\qquad\qquad (7" lb." +  8" oz.") + ( \bar{8}" lb."  +  4" oz."),`
`\qquad\qquad\qquad =  (7 - 8)" lb."  + (8 + 4)" oz."),`
`\qquad\qquad\qquad =  \bar{1}" lb." + 12" oz.",`

where the result is just a complementary form for `-4" oz."

His growth by today, on the other hand, is

`\qquad\qquad\qquad (8" lb." +  0" oz.") + ( \bar{8}" lb."  +  4" oz."),`
`\qquad\qquad\qquad =  (8 - 8)" lb."  + (0 + 4)" oz.),`
`\qquad\qquad\qquad =  0" lb." + 4" oz.".`

This method may look unnecessarily awkward for small and simple quantities such as we have here.  It is nevertheless extremely useful.  The usual method by which a computer represents a negative integer is as a complementary form in base two; the usual method of representing negative common logarithms uses complementary form; complementary form can be helpful in ordinary arithmetic.

Perhaps the most graphic example of complementary form mixes the old style mechanical odometer and the imagination.  But that is a topic for another post.