RogerBW's Blog

A Modest Proposal About Numbers 12 February 2015

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.


  1. Posted by chris at 12:22pm on 12 February 2015

    As with Esperanto (for which read any other "universal language" as well) or a more rational keyboard design, the difficulty is that you have to have it already for a change to it to be practicable.

  2. Posted by RogerBW at 12:26pm on 12 February 2015

    And how many people actually do mental arithmetic these days anyway?

    It would be easy to add this as a numerical display format on a reasonably open computer system, but the whole point of it is that it's useful when you don't have a computer to hand.

  3. Posted by Michael Cule at 12:26pm on 12 February 2015

    Ummm.

    No, I mean really Ummm....

    I can't tell whether this is meant to be... ummm....

    I try to think about explaining negative numbers to a five year old. "You have some beans... and you have some anti-beans..."

    (Chris! Chris! Could you check on Roger's temperature?)

  4. Posted by RogerBW at 12:28pm on 12 February 2015

    Meh, five is the best age, before they've been forced into the rigidity of conventional schools and the One True Way of doing things. Young kids are just fine with negative numbers if they haven't already been indoctrinated into thinking of numbers only as tools for counting with.

    Except they now start school at four or three. But that's another rant for another time, and I'll probably leave it to Chris.

  5. Posted by Michael Cule at 04:01pm on 12 February 2015

    Well, tell you what, first try to persuade the human race to start using base 12...

  6. Posted by RogerBW at 04:27pm on 12 February 2015

    Base eleven would make this system work much better.

  7. Posted by Michael Cule at 07:17pm on 12 February 2015

    /me facepalms...

  8. Posted by Chris at 08:55pm on 12 February 2015

    Oh please. Be real! Not a blasted prime.

    Ten is quite bad enough, only divisible by 2 and 5 instead of 2,3,4 and 6.

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