Why Is Multiplication in Natural Numbers Commutative?
Mathematics can be a tricky topic for many students, especially when it comes to the different properties required to understand why math works the way that it does. Having these properties broken down and explained can help to alleviate the confusion associated with them. One such complicated property is why the multiplication of natural numbers is commutative.
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Read Through the Multiplication Property
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The Commutative Property of Multiplication states for any two natural numbers (i.e. whole, positive integers such as 1, 2, 3, and so on), A and B, A x B = B x A. One example of this, using the numbers 3 and 7, is that 3 x 7 = 7 x 3, as 3 x 7 = 21 and also 7 x 3 = 21. Try this for several different combinations of natural numbers to test for yourself and to understand the process.
Recognize the Meaning of Your Findings
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The property seems confusing, but recognize what your tests are showing to you. A x B = B x A simply means that regardless of the order of the two natural numbers, multiplying them in that order will give you the same result.
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Give Yourself a Visual to Help
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Use a simple math problem and take time to lay out items like coins or small candies to help understand how the Commutative Property works. For the problem 3 x 7 = 7 x 3, lay out coins in two grids, one grid 3 coins wide by 7 coins long and one grid 7 coins wide by 3 coins long. Notice the shapes are the same, and if one grid was rotated, they would be identical.
A Math Proof to Prove the Point
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According to a proof carried out by educators in Utah, the following (greatly simplified) proof proves that multiplication is commutative:
Think of "m" as a fixed quantity. For N x M = M x N. Now assume M x 1 = M. Our goal is to prove if M x 1 = M, then 1 x M = M. Step 1 is 1 x 1 = 1. Step 2 is 1 x (N + 1) = 1 x N + 1 = N + 1. Steps 1 and 2 prove that 1 x N = N, for all numbers designated by N, including M. So, M x N = N x M is the same as (M x N) + M = (N x M) + M. Therefore, M x N = N x M.
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