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 ${\displaystyle 999.9}$.  Let's ignore the decimal point, and work in distance units of a tenth of a mile.  Then the trip meter reads ${\displaystyle 9999}$ units.  What it should read is ${\displaystyle -1}$ unit, or some equivalent to that.

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

We can put this in wiggle, and calculate

${\displaystyle \overline{1}999 = {\overline{1}}_{10}{9}_{10}{9}_{10}9,}$
${\displaystyle = {\overline{1}}_{1000}999,}$
${\displaystyle = -1000+999,}$
${\displaystyle = -1.}$

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

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

${\displaystyle ...999 = -1,}$

${\displaystyle ...998 = -2,}$

${\displaystyle ...997 = -3,}$

${\displaystyle ⋮}$