NavList:

Bill wrote

 I mention this as it puzzled me when I
was small if there was an alternative to negative numbers, some
quantity that did not obey the usual rules like -1 times -1 =1.

 Like you Like you I grew up when I used logs (and a slide rule) because there were no pocket calculators. I never really understood the logs of numbers less than one. After a quick bit of research involving the explanation pages in Nories tables AND a pocket calculator I think I have an explanation. Essentialy there are three ways of expressing the logs of numbers less than 1.  They are:

Mathematical

Bar

Tabular

First I use my fx-82 AU to determine the mathematical version

log 53.75  =   1.73062

log 5.378  =  0.73062

log 0.5378 =  - 0.2694  note change in mantissa

log 0.05378 = - 1.2694

log 0.005378 -  - 2.6294

When the number becomes less than 1 not only does the log become negative but the mantissa changes. This is confusing - numbers >1 have a different mantissa to numbers  > 1.  To quote Norie "it is the custom to express the log of a number less than 1  with a negative characteristic and a positive mantissa".

Continuing to quote Norie

 -2.2694  = -3 + 1 - 0.2694

                       = -3  + 0.7306

which is written as  bar 3 .7306

The mantissa does not change when the number is less than 1.   Note that  bar 3 .7306 is not a negative number. It has negative characteristic with a positve mantissa.

But I can think of another problem. How many bar symbols will the printer need to store and set? The solution in navigational tables is to create tabular logs by adding 10 to the characteristic. Thus

bar 3.7306 becomes 7.7306.

Knott's Four figure Mathematical Tables ( revised by Comrie, a mathemitician)  uses the bar notation but Norie and  Inman (navigation) use the tabular version. I also looked at Chambers Seven Figure mathematical Tables and became thorougly confused - I will not investigate further!

I hope that the above makes sense.......

I must open Libre office and Identify the  symbol set for bar notation.