Commutation of Addition is the rule that says “a+b = b+a”.
Commutation of Addition is an axiom. We don’t claim that we can prove it from our collection of starting rules, rather, it is one of the rules in our collection of starting rules.
However, we at Mockingbird Academy maintain that there is a way to visualize why commutation of addition is true.
Assume we spend a lot of money to put gold marks on an expensive stone tablet, and we end up with this:
As we analyze it, first we notice there are eight markings on it (we had to pay for each one) and we wanted to make sure we got what we paid for. Now for the magic–can you look at the tablet and see that it says “three plus five”? We can’t force you to see it, you are free in your own mind. If yes, then look at the next picture of the same tablet:
Using the same thinking as used for the previous view, can you see the above as representing “five plus three”?
We still have eight markings, we paid a lot of money so we wouldn’t have to worry about the number of markings changing when we slowly and carefully move the tablet.
3+5 = 5+3
If we think about what we have done here, we should be able to agree that if our strategy is to represent the addition a+b by two groupings of markings on a tablet, we can rotate the table by 180 degrees–as was done above–and we will have b+a.
a+b = b+a
Similar work has been done for Commutation of Multiplication.
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