HP Forums

Since we can't comment articles, I post a comment here: I believe floating point multiplication *is* commutative, but it is not associative. Same for +.

In the article Joe does acknowledge that multiplication is commutative, but the point he is trying to make is that because the resolution of a calculator is limited, having some round off error is unavoidable and because of that, changing the order of the terms may cause the result to change.

Edit: Now that I go and look, (school was a long time ago), addition and multiplication are also associative but again in multiplication rounding error will trip you up on a calculator.

I think the title of Joe's article is a bit provocative but not precise because it is missing some kid of qualifier. "Multiplication is not commutative", it is not commutative in general or on the Moon or maybe on the Mars. In general things stay the same and multiplication is commutative but on calculators or on computers real and irrational numbers are represented as approximations so we should not be talking about not commutative multiplication but about error the floating point multiplication introduces on HP Prime or on any other computing device.

'Multiplication is commutative' means a*b is equal to b*a.
That holds for floating-point multiplications as well.
What Joe said was (a*b)*c is not necessarily equal to a*(b*c), which is not the same thing.
Werner

(01-31-2014 01:24 PM)Werner Wrote: [ -> ]'Multiplication is commutative' means a*b is equal to b*a.
That holds for floating-point multiplications as well.
What Joe said was (a*b)*c is not necessarily equal to a*(b*c), which is not the same thing.
Werner

The property you are describing is associative, so I found with a bit of quick research earlier, and yes multiplication is associative. (3x2)x5 = 3x(2x5) but it breaks on calculators because of the limits of precision and that I think is the point of Joe's article.

(01-31-2014 02:25 PM)Paul Berger (Canada) Wrote: [ -> ]

(01-31-2014 01:24 PM)Werner Wrote: [ -> ]'Multiplication is commutative' means a*b is equal to b*a.
That holds for floating-point multiplications as well.
What Joe said was (a*b)*c is not necessarily equal to a*(b*c), which is not the same thing.
Werner

The property you are describing is associative, so I found with a bit of quick research earlier, and yeas multiplication is associative. (3x2)x5 = 3x(2x5) but it breaks on calculators because of the limits of precision and that I think is the point of Joe's article.

...which suggests the title "Multiplication is not associative" is more fitting than "Multiplication is not commutative"

(01-31-2014 02:38 PM)Han Wrote: [ -> ]

(01-31-2014 02:25 PM)Paul Berger (Canada) Wrote: [ -> ]The property you are describing is associative, so I found with a bit of quick research earlier, and yes multiplication is associative. (3x2)x5 = 3x(2x5) but it breaks on calculators because of the limits of precision and that I think is the point of Joe's article.

...which suggests the title "Multiplication is not associative" is more fitting than "Multiplication is not commutative"

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.

(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.

I stand corrected reading more carefully I see that commutative property is defined as only applying to binary operations. However if that is in fact the true definition then what did Joe's example demonstrate? It not associative, as the definition of associative is that the order of the terms is not changed, just the order of operations, hence the brackets in the example.

I stand corrected. The title and article should say "associative", not "commutative". Thanks to everybody who caught that. A pertinent footnote has been appended to the article.

Quote:I think the title of Joe's article is a bit provocative but not precise...

Homiletic license. ;-) Intentionally provocative ambiguity is one of my favorite rhetorical devices. It keeps the audience awake and thinking. However, I'll try not to misidentify math properties in the future.