Exact Question: Invent a single binary operator $*$ such that for every real numbers $a$ and $b$, the operations $a + b$, $a - b$, $a \times b$, $a \div b$ can be created by applying $*$ (multiple times), starting with only $a$'s and $b$'s
From my interpretation, you can apply $*$ recursively some number of times with carefully selected parameters to produce the desired outcome.
I thought the operator should be a combination of $a - b$ and $a \cdot b^{-1}$ Since $-$ and $\div$ can produce $+$ and $\times$ respectively
Please do not tell me the full answer. Give me a hint to point me towards the right path
I'm not sure that this is possible. You wish to define a binary operator $*:\mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ that has some properties, but in particular you must define what $0*0$ is equal to. This means that however $*$ is defined in terms of the four fundamental operations, it cannot involve division, since $0/0$ is an undefined quantity. But if we cannot involve division in the definition, then there is no way to get back $1/2$ from any number of applications (and groupings) of $*$ to $1$ and $2$, since $\mathbb{Z}$ is a ring under $+$ and $\times$.
Where does that leave us? The definition of $*$ we seek must use an operation beyond the four fundamental operations, and still must be able to recreate the effects of all four under some number of applications (and groupings)... which just strikes me as impossible.