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Q&A

Why should a non-commutative operation even be called "multiplication"?

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As per my knowledge and what was taught in school,

$a\times b$ is $a$ times $b$ or $b$ times $a$

Obviously this is commutative as $a$ times $b$ and $b$ times $a$ are same thing. On the other hand there are multiplications like vector multiplication and matrix multiplication that are not commutative.

What does multiplication mean in general, for these? Or should they even be called multiplication?

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Usually (not always, but true for the examples of multiplication of real numbers and multiplication of square matrices), a binary operation that is called some variant on ‘multiplication’ is the multiplicative operation of some ring. That's the closest thing to a rigorous definition of multiplication ‘in general’ that you're likely to find. The ring axioms don't require multiplication to be commutative; they only require that it is associative, that it has an identity element (usually called ‘1’ or ‘unit’ or some variation), and that it distributes over whatever is being called ‘addition’.

Occasionally conflated with the concept of ‘multiplication’ is the concept of ‘product’, which often carries slightly different connotations. Products usually generalize the idea of forming pairs of things—the Cartesian product of two sets $A$ and $B$, for example, is written $A \times B$ and means the set of all pairs where the first entry in the pair is any element of $A$ and the second entry is any element of $B$. The connection to real number products should be evident; for finite sets, the cardinality of (number of elements in) the Cartesian product of two sets is the real number product of the cardinalities of each of the sets. But the Cartesian product is not, strictly speaking, an operation that can be used as the multiplicative operation in a ring, because it doesn't have a unit—you can take the Cartesian product of a single-element set with any other set and get a set with the same cardinality, but it will be a different set (a set of pairs instead of a set of not-pairs). The generalization of this concept is perhaps best formalized by the category theoretical product, if you dare venture into those woods.

Finally, there are lots of examples of things called products or multiplication in mathematics not because they conform rigorously to one of the above two concepts, but because they involve or are closely related to something that does. Multiplying a vector by a scalar, for example, is not a ring multiplication or a product-like operation, but since it involves doing ring multiplication between the scalar and the vector elements, it gets called multiplication itself. This is how most things called ‘multiplication’ actually got their names, since many of them predate formal ring theory and category theory.

If you are an aspiring mathematician, the thing to remember when encountering a new operation called ‘whatever multiplication’ is that it is its own thing, and you need to pay attention to its definition and properties and not carry too many preconceptions from anything else called ‘multiplication’. It might be associative or even commutative... but it might not. It might have an identity, but it might not. It might be a way of pairing things together, but it might not. It might involve a more familiar form of multiplication in its definition, but it might not. The axioms and logic of higher mathematics are (ideally) rigorous, but the terminology often isn't. Naming things is hard, and changing the names of things is harder!

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General comments (2 comments)
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In most (but not all) cases where you name an operation multiplication is when you have two binary operations, and one of them distributes over the other, but not the other way round. Then the operation that distributes over the other is called multiplication and the other one addition, in analogy to the real numbers where multiplication distributes over addition.

Given that in those cases, almost always the addition is commutative, but often the multiplication isn't, there is also another convention where if you have only one operation, this operation is usually called multiplication if it doesn't need to be commutative, and addition if it is required to be commutative. For example, in general groups, the group operation is called multiplication, while in abelian groups, it is usually called addition.

Moreover in most cases where an operation is called multiplication, multiplication of natural or of real numbers is a special case of that structure.

For example, multiplication of ordinal numbers is not commutative, but the natural numbers are a subset of the ordinal numbers, and the restriction of ordinal multiplication to those gives the multiplication of natural numbers (and therefore this restriction is also commutative).

Also multiplication of real or complex matrices reduces to the multiplication of real or complex numbers in the case of $1\times 1$ matrices (or more generally, in the case of multiples of the identity matrix).

Also multiplication of non-zero real numbers is a specific instance of group multiplication, and without the restriction to non-zero numbers, a specific instance of field multiplication.

The cross product of vectors does not have such a special case, but it clearly distributes over addition.

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