Final answer:
The commutative property in arithmetic, both for addition (A+B=B+A) and multiplication (AB=BA), indicates that the order of numbers does not impact the sum or product. While this is self-evident or analytic for addition, it is less obvious or synthetic for multiplication; nevertheless, the result remains the same regardless of order.
Step-by-step explanation:
The question essentially asks for clarification on the commutative property of addition and multiplication in arithmetic. The commutative property states that the order in which numbers are added or multiplied does not affect the result. With the example of counting pots of beer, A+B=B+A means that whether you add 30 pots from Adamen to 50 pots from Biluda or vice versa, the result is the same: 80 pots. The order of adding does not matter. This property is analytic because its truth is self-evident by the definition of addition.
Regarding multiplication, the question notes that realizing AB=BA (such as calculating the total pots if Adamen sells in lots of 3 and you buy 5 lots, compared to Biluda selling in lots of 5, but you buy 3 lots) is not immediately obvious and is synthetic. It's surprising because the process of buying a certain number of lots seems inherently ordered, yet the totals come out the same (5 lots of 3 is 15 pots, and 3 lots of 5 is also 15 pots). This illustrates the commutative property for multiplication, which says that the product of two numbers is unaffected by the order in which they are multiplied.