Why do mathematicians use $\oplus$ instead of $+$?

Question

What is the historical reason for using $\oplus$ instead of $+$ to denote operations that are generally thought of as addition? Similarly, why is $\otimes$ used instead of $\times$ (or just $\cdot$) to denote operations generally thought of as multiplication?

Answer 1

These symbols have different meanings in different contexts. For instance, if we are talking about vector spaces then saying $V=U+W$ is different from $V=U\oplus W$

Answer 2

If $A$ and $B$ are modules over a ring, their direct product $A \times B$ and their tensor product $A \otimes B$ are different things, so it would be unhelpful to use the same notation for them.

Answer 3

Besides customary uses, I've seen it used to emphathize the differences of two types of addition, for example: $(\alpha \oplus \beta)A+B$.

The $\oplus$ refers to regular scalar addition, and the $+$ refers to matrix addition.