About Joe's article multiplication is not commutative

[Han]

(01-31-2014 02:49 PM)Paul Berger (Canada) Wrote:  Well from what I read if a function commutative it means you can change the order of the terms without changing the result, which is what Joe's example seems to be doing. My example above illustrates the associative property. I am not a maths expert by any means so if any of the maths teachers out wish to jump in feel free. I found my answers by googling "associative maths" and reading some of the results.

Joe's results actually indicate that the roundoff causes the associativity property to fail on calculators, whereas his title suggests that the commutative property fails when it actually holds. In order to demonstrate that the commutative property fails, his examples would need to show that \( a\cdot b \) is different from \( b\cdot a \). However, on a calculator, a product of two numbers will always be the same regardless of roundoff. On the other hand, a product of 3 values may differ slightly depending which two get multiplied first.