# What is zero times zero?

The standard answer `0*0 = 0` is fine for most people. But the odd person who found this page will probably have second thoughts about it. As a philosoper you may feel that ‘zero times nothing’ is even less than plain empty: a deeper void.
Shouldn’t we express this as a power `0^2` in mathematics? Or will all positive powers of zero necessarily collapse to `0` under the usual assumptions in arithmetic? Can’t we make an exception so that zero powers make sense?

You have to ask yourself: what is zero? There are two simple answers.
First answer is the +0 axiom that `a+0 = a`. The function of adding zero to a number is not to change that number. This is called the identity element of addition in field theory.
Second answer is that zero is defined as one, minus one. The 1- axiom substitutes a sign for a counted down entry `1- := 0` in my natural array system, where a number increments by unit `1` and the unit `-` decrements it.
Equality `X=Y` implies `Y=X` (a symmetry relation), but in substitution the right side replaces the left (not symmetric).

Under the usual field rules we have:
`0*0 = 0*0 + 0 = 0*0 + 1*0 = (0+1)*0 = 1*0 = 0`.
Using the +0 axiom, symmetry of equality, the identity element of multiplication `1*a = a` and the distributive law `(a+b)*c = a*c + b*c`.
By induction this reduces `0^n = 0` for all `n>0` in standard algebra.

If we’d like to maintain that `0*0` is somehow different from plain `0`, some exceptions must be introduced to the axioms of field theory. Simplest is not to let our +0 axiom apply to powers of zero. In turn we can discard the exception that there is no multiplicative inverse `1/0` and allow for negative powers of zero too.
Try on this zero division ring. If `0^-1 = 1.. = ω` continues to make sense, your arithmetical system just became rich!

With my 1- substitution axiom an expression with just zero values like `0*0` defies reduction altogether.
`1- + -*0 = 1 + -*1 + -*0 = 1 + -*(1+0) = 1 + -*1 = 1- := 0`.
`1- + -*0 := 0 + -*0 = 1*0 + -*0 = 1-*0 := 0*0`.
One way substitution to define inverses, how can we cope?

I can’t prove that `0+0 := 0` either, should I leave it, or make it a new rule?
`1- + 0 := 0 + 0 = 0*1 + 0*1 = 0*(1+1) = 0*2`.
`1- + 0 = 1 + -*1 + 0 = 1 + 0 + -*1 = 1 + -*1 = 1- := 0`.
To hold on to zero powers I maintain that `-*0 ≠ 0` and take `-*-` to be a double negated variation of `1` that also may raise special powers.

Some 10 years ago on my xs4all blog I wrote a series of articles, where I derived theorems from such a contrived substitution axiom. But I am sceptical now. To drive the rule of additive inverses into a one way street raises a serious obstruction in the centre of mathematics.
What do you think?

## 2 thoughts on “What is zero times zero?”

1. Dat ziet er goed uit! Nu wachten op reacties!

Kusjes Wattson

PS Mag ik hem “liken”?

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1. Je bent geabonneerd zeker? Ik was net aan het schrijven, maar Like! maar Wattson 😉

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