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While I know that both addition and multiplication are commutative operations, I can easily visualize that, e.g. 3 + 4 = 4 + 3 = 7 by thinking of seven objects in a row and separating them into two heaps of three and four objects, respectively. Now it's trivial to conclude that commutativity of addition amounts to labeling which one of the heaps is first and whics one is the second. This is arbitrary, therefore addition is commutative.

However, I cannot find a similar visualization to help me see why 3 x 4 = 4 x 3 = 12. I'm sure there must be a way.

  • How about viewing the product $a·b$ as calculating the area of a rectangle with corresponding side lengths $a$ and $b$, and then viewing commutativity $a·b = b·a$ in that context as the fact that its area doesn’t change with a reflection of the rectangle? – k.stm Sep 07 '14 at 23:15
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    Here's how I think about it. If we have 12 objects, we can arrange them in a lattice that is 4 rows by 3 columns (4 x 3 = 12). We can also transpose that lattice so it is 3 rows by 4 columns (3 x 4 = 12). – Nishrito Sep 07 '14 at 23:16
  • See my answer to this question. – Lucian Sep 07 '14 at 23:20

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