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?

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    $\begingroup$ So you won't confuse them with $+$ and $\times$? $\endgroup$
    – Asaf Karagila
    Commented Jul 26, 2015 at 0:21
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    $\begingroup$ 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
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    $\begingroup$ The symbol $\oplus$ is not synonymous with $+$. ${}\qquad{}$ $\endgroup$ Commented Jul 26, 2015 at 0:22
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    $\begingroup$ @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
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    $\begingroup$ @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


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.


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 $V=U\oplus W$

  • $\begingroup$ 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

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.


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