The question of whether or not a given operation of commutative is, at first, a remarkably simple one to answer, especially when considering the most basic mathematical operations.
Addition, for example, is certainly commutative, as any basic addition problem can be reversed with absolutely no change in the equation's outcome: 4 + 3 = 7 just as 3 + 4 = 7.
Subtraction, however, is most definitely non-commutative: 4 - 3 = 1 whereas 3 - 4 = -1.
Multiplication is commutative for the same reason as addition: 4 x 3 and 3 x 4 both equal 12; whereas division is non-commutative because 4/3 does not remotely equal 3/4.
These four most basic operations can be combined and rearranged at will, but the commutative properties of the various operators will certainly remain true, making commutativity a rather valuable tool even for those who only desire to know the most basic principles of mathematics.
Commutativity Beyond the Basics
Moving beyond mere addition, subtraction, multiplication, and division, however, the world of commutativity becomes far murkier, for here, even an operator such as division can become non-commutative.
Matrix multiplication, for example, is actually non-commutative, as multiplying two matrices together is fully dependent upon the order in which the operations are performed. Therefore, reversing or changing the order in any way dramatically alters the results of the operation as a whole.
Similarly, in the mathematics of set theory or group theory, commutativity can most certainly be altered depending on the nature of the set being operated upon.
There is also an entire branch of mathematics known as Non-Commutative Geometry, which deals with the non-commutativity of operations performed on complex structures such as manifolds and other topological spaces, a topic which holds many applications, both within pure non-Euclidean mathematics as well as within mathematical physics.
Non-Commutativity in Nature
Moving briefly beyond the realm of pure mathematics, however, it is rather easy to see examples of commutativity and non-commutativity in every area of life and the natural world.
A most simplistic example might be the eating of an ice cream cone. Is this activity commutative? In order to determine this one need only ask themselves whether it matters which is eaten first - the ice cream, or the cone? Clearly this represents a commutative activity, for eating the cone first most assuredly changes the entire course of the eating process.
On the other hand, what about putting on a pair of socks? Does it matter in which order the socks are put on one's feet? Probably not. Therefore, this is a commutative activity.
Now, beyond these rather simple activities (and the countless other commutative and non-commutative events which most certainly occur rather frequently without notice), a little thought can turn up an entire host of other problems of commutativity just outside this "common" realm.
In physics, for example, commutativity has played a rather massive role. Quantum mechanics, for example, has found some of its most dramatic experimental results by demonstrating rather counter-intuitive forms of non-commutativity. This concept lies at the heart of Heisenberg's famous uncertainty principles (which makes sense because this principle is founded upon the mathematics of matrices, which, as has already been stated, are non-commutative).
Similarly, the question has been raised by many scientists whether there are operators within the human brain itself which are inherently non-commutative, an idea which may begin to explain some of the subtleties of human consciousness.
Clearly, when one moves beyond the simplicity of elementary mathematics, this seemingly tame and simple concept of commutativity becomes both vitally important, and absolutely fascinating.
Isaac M. McPhee - Isaac McPhee was born as a human child in Mt. Vernon, WA, c. 1982; he currently resides in the bustling heart of New York City where he ...
