## Saturday, November 24, 2012

### Abolishing zeros

I've just finished a short paper with the title 'Accounting: counting with balanced Debits and Credits' The paper aims to eradicate the notion of zero and thus division by zero. Here is the abstract:

Division by zero is undefined, so we have decided to eradicate zeros completely. We do this by introducing a new kind of Integer, called an Account, which is an ordered pair of two Naturals - Debit and Credit.
Naturally, we also introduce a new kind of Rational, called the Super-Rational, which is an ordered pair of two Accounts - Numerator and Denominator.

1. Great! Now multiplying x by 0 (or rather its equivalent because we have a zero but we don't call it zero any more) is x! And I always thought it would be 0. But bankers think differently of course. Will advice my bank to use the new rational system.

2. No, I think you rushed your accounting!

Let's say that x is represented by Account x+1\1 and then you multiply(**) that with a balanced Account y\y.

Following the definitions of ** we get the following result ((x+1)*y+y)\((x+1)*y+y) which is still a balanced Account.

I really do think that bankers would like that. In fact: I work at a bank!

3. > In fact: I work at a bank!
I know. That's why I made the allusion. ;-)

But now humor aside:
**: a\b ** c\d => ((a*c)+(b*d)\((a*d)+(b*c))

So let's say your first factor is 1\1 which is equivalent of zero then all a's and b's in the defintion cancel out leaving
(c+d)\(d+c) which is again zero. So zero multiplied by something is zero as it should be. But what has been gained? Replacing zero (0) by an endless number of zeros (a\a)?

The real litmus test is division. Zero divided by anything gives zero and division by zero also gives zero. I agree, we should have always allowed division by zero and called the result zero. No information would have been lost.

4. > I know. That's why I made the allusion. ;-)
That's cheeky!

You are still calculating with zeros. Please do not use the word zero anymore!
Replace them with balanced Accounts of Debits and Credits and operations on them.

Anyway, let's do the multiplicative inverse of a (numerated) balanced Account. My paper already has given a glimpse of that:

`1\1/1\2 == 1\2/1\1

Is 1\2/1\1 in the same equivalence class as your zero? I don't think so.

5. I use zero for the zero equivalent class in your system and for 0 in the "common" system to make things comparable. I just would like to understand what the advantage of your system is. Can I get rid of singularities in physics because there is no division by zero any more?

Could you confirm this sentence:
> Zero divided by anything gives zero and division by zero also gives zero.

6. Your statement: "Zero divided by anything gives zero and division by zero also gives zero." is not true.
Also, there are no singularities but only different equivalent classes that are incomparable. A number that is 'divided' by 'zero' puts that number in a different equivalence class - let's call it 'infinity'.
Unfortunately, it is impossible to recover from 'infinity' back to the original number. The same applies for 'zero' 'divided' by 'zero' which is an equivalence class you cannot escape. Whatever you do: 0/0 will remain 0/0.
There is a way out but I leave that for a future post!

7. Will wait for it.

9. ... so they get a paper out of renaming the constituents of the standard construction of the integers and the standard construction of the rationals. Wooow.

10. But now based on lazy evaluation (don't simplify), instead of strict evaluation (always simplified). That makes a *big* difference. Sorry that I did not make that clear.