Which operations are not commutative

Subtraction, division, and exponentiation are not. A useful example of associative operations is the composition of functions. Usual sum and multiplications are commutative. Matrix product is not commutative, the composition of functions is not commutative in general a rotation composed with a translation is not the same as a translation first, and then the rotation , vector product is not commutative.

Non commutativity of composition of functions is on the basis of Heisemberg uncertainty principle. The Hamilton Quaternions are a useful structure used in math. They have a multiplication which is associative but non commutative. Such properties are important because they make an operation user-friendly. This seems kind of obviuos, but it depends on the properties of the operation. Concluding, I would say that when a mathematician hear the word multiplication , immediately think to an associative operation, usually but not always with neutral element, sometimes commutative.

That is a theorem not a definition , thank you very much. Of course the easiest proof is geometric: the former is an a x b rectangle, and you rotate it and you get a b x a rectangle.

To prove it from axioms is a bit trickier because you have to answer the question if multiplication is not commutative as an axiom then what do we prove it from? The answer is some combination of mathematical induction, which is the idea that if something works for all numbers "and so on" then it works for all numbers, but it only applies in the integers because other number systems don't really have an "and so on.

So once you talk about multiplication outside of the integers you lose all these good assurances that multiplication is commutative. In my opinion, that's really it. Multiplication makes sense in a lot of places, and many of those places don't have strong reasons that it would be commutative. It ought to be mentioned that multiplication in general is an action of a collection of transformations on a collection of objects, and multiplication of elements of a group is in fact a special case.

This might have even been inherent in the original notion of multiplication; consider:. The latter makes no sense. What is happening here? Even in many natural languages, numbers have been used in non-commutative syntax in English, "three chickens" is correct while "chickens three" is wrong , because the multiplier and the multiplied are of different types one is a countable noun phrase while the other is like a determiner.

English requires some details like pluralization and articles, but mathematically the multiplication is as stated. Observe that multiplication between natural numbers is commutative, but multiplication of natural numbers to countable nouns is not. So for example we have:. And in mathematics you see the same kind of scalar multiplication in vector spaces, where again the term "scalar" related to "scale" is not accidental.

Again, multiplication of scalings to vectors is not commutative, but multiplication between scalings which is simply composition is commutative. More generally, even the transformations that act on the collection of objects may not have commutative composition.

But maybe the "multiplication" name still stuck on as it did for numbers and scalings. That is why we have square matrix multiplication defined precisely so that it is equivalent to their composition when acting on the vectors. Commutativity is not a fundamental part of multiplication. It is just a consequence of how it works in certain situations. Here is an analogy. When you multiply two positive integers, a and b, you can define multiplication as repetitive additions.

This is how multiplication is taught in the early years. Now have a look what happens when we use this on negative integers. Nothing strange so far. But we will repeat this -3 times. Hold on, how can we do something a negative amount of times? Why should we call this multiplication when it is not a repetitive addition? My point here is that when we transitioned from just positive integers to include the negative integers, we lost something that we thought was fundamental.

But it was never fundamental. Let's have a look at matrix multiplication. For instance, one can think of a translation of axes in the coordinate plane as an "element," and following one translation by another as a "product. This operation is commutative. If the set of transformations includes both translations and rotations, however, then the operation loses its commutativity. A rotation of axes followed by a translation does not have the same effect on the ultimate position of the axes as the same translation followed by the same rotation.

To figure out if there is a simple way to tell if a set with an operation table is commutative under the given operation,. Do we notice a pattern here in the tables when a set is commutative under a certain operation? Notice that for this table, each result is exactly the same as the one that mirrors it on the other side of the diagonal; if we were to fold this table in half right on the diagonal the purple line then the two folded sides would be exactly the same : the b would fold right on top of the b on the other side of the diagonal, the c would fold right on top of the c right on the other side of the diagonal and the c would fold right on top of the c on the other side of the diagonal!

The two halves of results on either side of the diagonal mirror one another, so this operation table is symmetric across the diagonal!