Properties of real numbers

J

John Whitaker

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Is there a (non-confusing) way to distinguish between the Associative property and the Commutative property? No matter how I write either one, they both come out the same. Why do we need both Associative and Commutative properties? Thank you. John Whitaker
 
Well they are so clearly different it is hard to confuse the two.

The commutative property involves two elements in the set: \(\displaystyle a \circ b = b \circ a.\)

Whereas the associative property involves three elements in the set:
\(\displaystyle \left( {a \circ b} \right) \circ c = a \circ \left( {b \circ c} \right).\)

Above \(\displaystyle \circ\) represents whatever operation under discussion.


To complete the answer to your question; it is not a matter of need.
The two serve different functions literally.
The operation of matrix multiplication over the set of \(\displaystyle n \times n\) matrices is associative and not commutative.
 
Thank you. I understand the Commutative explanation, but what are the 3 elements you refer to in the Associative example? I see only 2 separated by an equal sign.
John Whitaker
 
The elements are the a, b, &c!
Operations are done on elements.

Commutative says that a operating b is the same as b operating on a.

Associative says the output on a operating on b operates on c is the same as a operating on the output on b operating on c.
 
As I understamd it now: Commutative involves only 2 elements; Associative involves 3 or more; but, Commutative is always 2.

Why couldn't the demented author of my textbook explain it as simply as you guys did?

Thank you, pka and Denis.
John Whitaker
 
John Whitaker said:
Commutative involves only 2 elements; Associative involves 3 or more; but, Commutative is always 2.
Click to expand...
Not exactly. The Commutative Property is what justifies saying that "2 + 3 + 4" equals "2 + 4 + 3". And the expressions each have three terms.

To "commute", in a sense, means to move. To "associate", in a sense, means to group together.

The Commutative Property refers to the ability to move terms around. The Associative Property refers to the ability to re-group terms.

Eliz.
 
stapel said:
The Commutative Property is what justifies saying that "2 + 3 + 4" equals "2 + 4 + 3". And the expressions each have three terms.
To "commute", in a sense, means to move. To "associate", in a sense, means to group together.
Click to expand...
What in the world does to move mean in mathematics?
All operations are binary. Are they not?
 
What in the world does to move mean in mathematics?
All operations are binary. Are they not?
Click to expand...

I don't know, pka....

but you will surely agree that the commutative property of addition for the set of real numbers says that

a + b = b + a

The positions of "a" and "b" on the left side and on the right side are clearly DIFFERENT, are they not? Could one reasonably assume that "a" has been moved to where "b" was, and that "b" has been moved to where "a" was? I'm probably missing something.

I've always explained to students that the commutative property allowed one to switch the position of two operands with respect to the operator symbol (either + or x), and that the associative property allowed one to change the grouping of the operands (using + only, or x only)....it always worked. Was that just luck? Was I wrong in this explanation?

If you have an example that would prove my explanation incorrect, I'd be more than happy to see it.
 
stapel said:
To "commute", in a sense, means to move.
Click to expand...
pka said:
What in the world does to move mean in mathematics?
Click to expand...
In a crude sense, it means "to move the numbers about the page". For instance, in "2 + 3 = 3 + 2", I moved the 2 and the 3 around, reversing which was on the right and the left of the "plus" sign.

I'm not familiar with many struggling beginning-algebra students who are familiar with, let alone comfortable with, "binary operators over fields", so I was attempting to provide an intuitive and non-technical definition.

I apologize for any confusion.

Eliz.
 
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