What will $AB$ be when $A=2, B=3$? My teacher says 6, But I do not agree. Because there is no operator between $A$ and $B$. Now we are arguing.
I know that $AB$ means $A\times B$ in the daily sense. But it loses the rigorousness of mathematics.
So my question is that, in the rigorous mathematical sense, $AB$ is different from $A\times B$, Right? Because in group theory, you always have to specify your binary operator.
Notations and symbols are mostly understood with respect to the context.
If $A=2$ and $B=3$ then $AB$ can stand for $23$ which is not $6$.
So we have to pay attention to the context.
For example when we write $35$ we mean $3\times 10 +5$ and we do not mean $3\times 5$
When we deal with vectors, $A\times B$ means cross product and $A.B$ means dot product which are totally different concepts.
In group theory $AB$ and $BA$ are not necessarily the same element while in number theory they are the same number.
Pay attention to the context and everything starts making sense.