TITLE: Is there an numeric arithmetic with a single operator?
QUESTION [13 upvotes]: I remember reading a while back about arithmetics with only a single operator that did all the work of the familiar $+ - \times /$ operators. If there is such a thing, could someone point me to it?
As background: I have a side project where I'm playing with neural nets and genetic search algorithms. Right now it's the good old kind, with matrix multiplication and addition. I thought it would be interesting to implement a sort of neural network that used a combined or "all in one" operator.
I've googled like heck and can't pick up the trail of where I thought I saw that before.
Is there study in that area, or am I mis-remembering, or worse, just making it up? I'd appreciate any pointers.

REPLY [8 votes]: This discussion follows the solutions of problems 1–2 given in A Problem Seminar by Donald J. Newman (Springer, 1982), pp. 45–46.
First, note that all four of $+, -, \times, \div$ can be built just from $-$ and the reciprocal operation $x\mapsto \frac1x$: Addition is easy: $$x+y = x-((x-x)-y).$$
By partial fraction decomposition we have $$ \frac1{x-x^2} = \frac1{1-x} + \frac1x$$
So $$x^2 = x-\left(\frac1{1-x} + \frac1x\right)^{-1}.$$
Now we can calculate $(x,y)\mapsto -2xy$ by using $-2xy = (x-y)^2 - x^2 - y^2$.  We can get rid of the $-2$ by using $\frac u{-2} = \left(\left(0-\frac1u\right)-\frac1u\right)^{-1}$.  And now that we have multiplication, obviously we get $x\div y = x\cdot \frac 1y$. So we have all of $+,-,\times,\div$ from $-$ and reciprocal.
Second, we need to find a single operation that gives us $-$ and reciprocal.  Newman observes that nothing like $x\circ y = x(x+y)$ can work because it can never give us subtraction.  So instead, we try something like $$x\circ y = \frac1{x-y}$$ that has subtraction and reciprocal in it already. 
Reciprocal is easy: $$x\circ 0 = \frac1{x-0} = \frac1x.$$
And then $$(x\circ y)\circ 0 = \frac1{\frac1{x-y}-0} = x-y.$$
So all four of $+, -, \times, \div$ can be defined in terms of $\circ$.
Looking at this now, I notice that Newman has tacitly assumed that we are allowed to use certain constants: $0$ in the second part, and 1 in the first part. (0 in the first part can be synthesized from $x-x$.) I don't know if these can be avoided, but perhaps you don't care about that detail.