The Commutativity of Real Numbers
In the previous discussion, we have shown that addition of two integers a and b is commutative by considering a and b as lengths of two segments. Joining the two segments is the same as adding two integers. We have observed that a + b equals b + a.
We have also shown that multiplication of two integers a and b is commutative by considering a and b as side lengths of a rectangle. Getting the area of a rectangle with side lengths a and b is the same as multiplying a and b. We have observed that ab equals ba. However, we argued about commutativity that any segment does not change its length and the movement of rectangles does not change its area. This reasoning is only limited to positive integers because we cannot represent negative integers as length or area. To explore further, we use the addition and multiplication tables to see if the addition and multiplication of integers are both commutative.