A small change to our numbering system would make daily use of numbers
remarkably easier.
That is to replace the number symbols from zero to nine with a
set that runs from -5 to +4.
For clarity I will use (-5), (-4), …, (-1), (+0), (+1), …, (+4) to
indicate the symbols. (One of the sets of circled-number glyphs in
Unicode, probably the enclosed alphanumerics starting at U+2460, would
do nicely, but may not be readable for everyone: ⑤ ④ ③ ② ① 0 1 2 3 4.)
Basic counting from one to ten runs
(+1), (+2), (+3), (+4), (+1)(-5), (+1)(-4), (+1)(-3), (+1)(-2),
(+1)(-1), (+1)(+0).
Things get unexpected in the mid-forties:
(+4)(+3), (+4)(+4), (+1)(-5)(-5), (+1)(-5)(-4), …
As one approaches 100:
(+1)(+0)(-2), (+1)(+0)(-1), (+1)(+0)(+0), (+1)(+0)(+1), …
(While I invented this approach myself, I gather that Yoruba counting
does something similar. So does balanced ternary arithmetic.)
Addition becomes slightly more complex because it has to incorporate
the idea of negative numbers:
45 + 38 = (+5)(-5) + (+4)(-2)
(+5)(-5) +
(+4)(-2)
(-5) + (-2) is (-1)(+3), so the last digit is (+3), carry the (-1)
(+5) + (+4) + (-1) is (+1)(-2), so the answer is (+1)(-2)(+3)
which is indeed 83 in the old notation.
But subtraction is now just a special case of addition, so there's
only one procedure to be taught rather than two. Simply flip the sign
of the subtrahend (yes, I had to look that up) and add:
55 - 18 = (+1)(-4)(-5) - (+2)(-2)
which is (+1)(-4)(-5) + (-2)(+2)
(+1)(-4)(-5) +
(-2)(+2)
(-5) + (+2) = (-3)
(-4) + (-2) = (-1)(+4), carry the (-1)
(+1) + (-1) = (+0), so the answer is (+4)(-3), 37 in old notation.
(One small bug: flipping the sign of numbers including the digit (-5)
is not as trivial as for other digits. That's what we get for having
an even base for our numbering system.)
Long multiplication and division are performed as before.
The obvious advantage is in approximation: when rounding a number,
just make the final digits (+0) and you'll have rounded to the nearest
multiple of 10, 100, or whatever. (And "£x.99" prices would disappear
forever, even if they would immediately be replaced by "£x.(+4)(+4)".)
The system also works in other number bases, though the representation
of binary is conveniently unchanged.
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