Conjecture Division of whole numbers?

Question

My daughter asked me for help with her math homework and I don't even understand what the question means. You don't have to give the answer but could someone point us in the right direction for finding the answer? This is the question: State whether the conjecture division of whole numbers is commutative is true or false. If false provide a counter example. Thanks ahead of time.

Answer 1

To elements are said to commute if the following holds:

ab = ba

Example of this:

1 x 2 = 2 x 1 = 2

It doesn't matter in which order you multiply them because it will always yield the same result.

Now this is true for multiplication of the integers.  However, it is false for division because the order in which you divide is important:

3/4 = 0.75, but 4/3 = 1.333...

Therefore, the integers are not commutative under division.

Hope this helps!

EDIT:  I'll treat it in general:

For two elements to commute under division we must have that:

a/b = b/a

Which yields:

a² = b²

Taking the root of both sides we get that:

a = ± b

Therefore, for division to commute we must have that a = b or a = -b.

Two examples:

a = 3, b = 3

a/b = 1 and b/3 = 1

a = -2 and b = 2

a/b = -1 and b/a = -1

Hope this helps some more!

Answer 2

What that means is can the numbers be moved around (commuting!)

Does 2/3 = 3/2 ?  NO!  So the answer is false.

For addition and multiplication, commutative property is true.

For subtraction and divistion it's false.

Take care,

David