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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