2
$\begingroup$

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?

$\endgroup$
8
  • 10
    $\begingroup$ So you won't confuse them with $+$ and $\times$? $\endgroup$
    – Asaf Karagila
    Commented Jul 26, 2015 at 0:21
  • 6
    $\begingroup$ In general, $\otimes$ and $\oplus$ are not user for such things. They are rather used for operations on algebraic objects such as rings, groups, algebras and modules. $\endgroup$
    – Pedro
    Commented Jul 26, 2015 at 0:22
  • 9
    $\begingroup$ The symbol $\oplus$ is not synonymous with $+$. ${}\qquad{}$ $\endgroup$ Commented Jul 26, 2015 at 0:22
  • 4
    $\begingroup$ @nathey: Sometimes we do overload those symbols. Other times it is important to make a distinction. This is not a mysterious issue, it is a simple question of people writing in a way that communicates their meaning. $\endgroup$ Commented Jul 26, 2015 at 0:30
  • 2
    $\begingroup$ @nathey: They did. Then they come up with some different operations on algebraic objects that were also analogous to addition and multiplication, and needed new symbols to distinguish them. $\endgroup$ Commented Jul 26, 2015 at 0:31

3 Answers 3

14
$\begingroup$

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.

$\endgroup$
11
$\begingroup$

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 $V=U\oplus W$

$\endgroup$
3
  • $\begingroup$ Oh yeah, at least one case where $+$ and $\oplus$ coexist. $\endgroup$ Commented Jul 26, 2015 at 0:29
  • $\begingroup$ I'm assuming that was meant to be condescending. Can you give me an instance you speak of where $\oplus$ is used for something "generally thought of as addition"? $\endgroup$ Commented Jul 26, 2015 at 2:45
  • $\begingroup$ $\oplus$ isn't used anywhere for addition, but the question is why shouldn't it be? The answer is because of places where the two clash like this. $\endgroup$ Commented Jul 29, 2015 at 6:21
0
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .