Ok, this node explains an unusual way of representing numbers, that has some advantages, but looks quite strange initially. I hope you find it fun; but it's probably for hardcore maths nuts only ;-)

metanote- this node uses some 'pre' formatting- there's no other way to get bars here.

This decimal number representation scheme is based on a per-digit bar notation that indicates whether that digit is negative. The big advantage it has is that you only need to know how to multiply the numbers from 1 to 5 to be able to do normal decimal arithmetic.

All numbers in this notation go between -5 and 5:

_ _ _ _ _                       
5 4 3 2 1 0 1 2 3 4 5 

If you want 6, you carry to the tens column:
      _                 __                   _ _
i.e. 14,   Minus 43 is: 43   and minus 56 is 144
Adding/subtracting is fairly obvious:
 _        _
14 + 1 = 13

          _
10 - 4 = 14
Carry works like this:
          _
15 + 1 = 24

And:
 _        _
14 - 1 = 15 = 5
                                                      _
i.e. you take from the tens column when you wrap from 5 to 5
and vice versa- if you go above 5 you add 1 onto the tens and 
go to minus 4.
                                    _
(Incidentally you try to get rid of 5 by convention)

So far, fairly easy ;-), if not read it again and try a few examples.

Let's try multiplying:

The total multiplication times table is:


   1  2  3  4  5
----------------
1 |1  2  3  4  5
  |      _  _
2 |2  4 14 12 10
  |   _  _
3 |3 14 11 12 15
  |   _     _
4 |4 12 12 24 20
  |
5 |5 10 15 20 25
So you only need to know 1/4 of the multiplication table.

Anyone can remember their five times table! ;-)

You can easily get the other 3/4 of the normal multiplication
table using the bar notation to your advantage:
                       ___
         _   _____       _     _
e.g. 2 * 3 = 2 * 3  =  1 4  =  1 4  (= -6)

Ok, so we should try long multiplication:

      2 4

      4 5 *
  -----------
    _
  1 2            (2 * 4)
      _
    2 4          (4 * 4)

    1 0          (2 * 5)

      2 0        (4 * 5)
--------------
      _
  1 1 2 0

 = 1080 in normal numbers which is correct.

You can also do division, subtraction and so on with this idea.

You can extend this idea to other bases, and as you probably already know your 8 times table, you can do multiplication in hexadecimal; it's still hard though.

p.s. I read this in a magazine of some kind; might be Scientific American; more than a decade ago; anyone?

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