The reversed odometer

For simplicity, consider a vehicle with an old-style mechanical odometer, and a separate trip odometer with four digits, the lowest digit representing tenths of miles.  Reset the trip odometer so that it reads all zeroes, and drive a tenth of a mile backwards.  The odometer should now read `999.9`.  Let's ignore the decimal point, and work in distance units of a tenth of a mile.  Then the trip meter reads `9999` units.  What it should read is `-1` unit, or some equivalent to that.

It should be clear that the trip meter, if unreset, cycles after every `10000` units of travel.  It could measure the actual distance since it was last reset, or it could be any multiple of `10000` units out.  Anyway, if the thousands digit reads `9`, but the meter is too high by `10000` units, then to correct the problem, we can simply remove `10` thousand from `9` thousand.  This gives `-1` thousand, or `\bar{1}` thousand---i.e. the thousand digit might say `9` but it will be better for us to read it as `\bar{1}`.

We can put this in wiggle, and calculate

`\qquad\qquad\qquad \bar{1}999  =  \bar{1}_{10}9_{10}9_{10}9,`
`\qquad\qquad\qquad\qquad  =  \bar{1}_{1000}999,`
`\qquad\qquad\qquad\qquad  =  -1000 + 999,`
`\qquad\qquad\qquad\qquad  =  -1.`

Put another way, provided we read the high order digit in suitable way, as string of nines, or else a `-1` followed by a string of nines, can be reasonable way to express `-1`.

This particular notation for negatives is called tens complement.  In tens complement, as for the odometer driven in reverse :

`\qquad\qquad\qquad \ldots999  =  -1,`

`\qquad\qquad\qquad \ldots998  =  -2,`

`\qquad\qquad\qquad \ldots997  =  -3,`

`\qquad\qquad\qquad\qquad\qquad\qquad \vdots`