Encoding 2 numbers into 1 Say we have two integers, $a$ and $b$. I need a way to combine these numbers into one unique number $x$, such that they can both be recovered from $x$ and no other numbers can be recovered from $x$
There are plenty of ways to do this, but the problem is that I am using this for programming and I only have the following operators:
$+$ $-$ $\times$ $\div$ $a^x$ $\sqrt{}$
I also have sin and cos but I doubt those will be needed. 
Is it possible to encode and decode $a$ and $b$ using just these operators?

Try $x=2^a (2b+1)$.
This relies on every integer being written uniquely as a power of $2$ times an odd number.
It is easy to recover $a$ and $b$: just divide $x$ by $2$ until you reach an odd number. Then $a$ is the number of divisions and $b$ is easily extracted from what is left.
However, if you need this for programming, then $a$ will be constrained to be quite small, if you use native numbers. In this context, the problem cannot be solved in full generality because you cannot compress all $2n$-bit strings into $n$-bit strings injectively.
