Why is multiplication a commutative operation?

Question

This trivial question is all about reasoning (intuition) and obviously not proving. I know $a\cdot b = b\cdot a$ from very early school years and it's considered intuitive. A simple proof is by taking a rectangle that is $2 \cdot 7$ and calculate the area which is $14$. The same is true if we rotate the rectangle. That however is a just the proof, it just does not explain the intuition behind this trivial theorem.

But, how can you say that when we add up seven twos, the result is equal to adding up two sevens?

Edit: It's about the why and not how and in fact I think it needs a bit of philosophical answer.

Answer 1

This is not a rigorous proof but I can give sort of an intuition.

It's because addition is commutative and Every Natural number comparatively larger can be represented as sum of two or more smaller numbers. for example:

$2 \cdot 4$ basically means adding two four times, i.e $2+2+2+2$ which turns out to be $8$: $$2 \cdot 4 = 2+2+2+2 = 8,$$ whereas $4 \cdot 2$ means adding four two times, i.e $4+4$ BUT $4 = 2+2$. Therefore $4+4$ becomes $(2+2) + (2+2)$. Since addition is commutative I can remove the brackets and it becomes $2+2+2+2$ which is equal to $8$: $$4\cdot 2 = 4+4 = (2+2)+(2+2) = 2+2+2+2 = 8.$$ Hence multiplication is commutative. And if you ask a bit more like why addition is commutative it's what seems so at least from our human reasoning and observation. Another reason I have stated this is precisely why division ain't commutative – because subtraction is not commutative.

Answer 2

Suppose that you have $7$ red balls, numbered from $1$ to $7$, and $7$ green balls, also numbered from $1$ to $7$. If you pick the red balls and then you pick the green ones, in the end you will have $2\times7$ balls. And if you pick first the balls with the number $1$, then the balls with the number $2$, and so on, in the end you wil have $7\times2$ balls. But its the same set of balls. Therefore, $2\times7=7\times2$.

Answer 3

The proof that the natural numbers is commutative is the “why”. Constructive proof of something is the closest thing to an objective “why” that we can get. Mathematics does not admit motivations and aspirations like a human agent might — there may very well be no deeper “why” as to why this is the case. Certain structures just have intrinsic properties.

Here is the proof that integer multiplication is commutative. It’s the best answer you’re going to get as to “why” this is the case. There may very well not be some deeper philosophical answer, and I suspect that any attempt to find one would be itself a bit of a philosophically unsound endeavour.

Answer 4

In mathematics, one does not discover rules that already exist. One makes up rules, and if something interesting comes out of those rules, you keep them. The made-up rules are called axioms. What comes out of them is a theorem.

In most cases, the commutativity of multiplication is an axiom. We keep it around because all of arithmetic, algebra, trigonometry, calculus, differential equations, etc., is based around this rule, so it is very interesting.

Answer 5

It seems that you are referring to multiplication in $\mathbb R$.

The geometric answer to this question is

Multiplication in a division ring $D$ is commutative precisely when the plane $D\times D$ satisfies the theorem of Pappus.

So in principle, you can take $\mathbb R\times \mathbb R$ and "forget" that commutativity works, and then use the fact that Pappus' theorem holds to deduce that multiplication is commutative.

I think these books at least contain proofs

Artin, Emil. Geometric algebra. Courier Dover Publications, 2016.

Kaplansky, Irving. Linear algebra and geometry: a second course. Courier Corporation, 2003.

Hartshorne, Robin. Geometry: Euclid and beyond. Springer Science & Business Media, 2013.

In Hilbert, David. The foundations of geometry. Open court publishing Company, 1902. Hilbert proves that real multiplication is commutative because Pascal's theorem holds in the real plane, but I think the idea is that you also need the division ring to be Archimedian to imply that it also satisfies Pappus' theorem.

This information is sort of hard to find because so often we axiomatize $\mathbb R$ by assuming commutativity. In contrast, the above books discuss constructing the real line and real plane with geometric axioms, and then one can build an "algebra of segments" which turns out to be nothing more or less than $\mathbb R$.