About the nature of the associative property ?

Question

The Associative Properties of
Addition and Multiplication are like this it says:
a + (b + c) = (a + b) + c and a(bc) = (ab)c
QUESTION: could you also write them like this
a + (b + c) = (c + b) + a and a(bc) = (ca)b

and Is there an associative property of division also ?

Answer 1

To your first question the answer is no. The way you rewrite the equations requires the commutative law as well as the associative law, because you switched the order of the things you were adding or multiplying. The associative law deals with how you group the numbers before adding or multiplying, but it doesn't allow you to change the order of the numbers. The commutative law does that.



To your second question, division is not associative. Example: (64/4)/2 = 16/2 = 8; but 64/(4/2) = 64/2 = 32.



Subtraction is also not associative.

Answer 2

First question: The associative property of addition says, very specifically, that for all numbers a, b, and c, it's true that a + (b + c) = (a + b) + c.



The associative property of multiplication says, very specifically, that for all numbers a, b, and c, it's true that a(bc) = (ab)c.



So the two equations you wrote in your question are true for real numbers, but they are not examples of associative properties.



The first equation in your question actually follows from just the commutative property of addition:



a + (b + c) = (b + c) + a [commutative property of addition--we switched the order of a and (b + c) around a plus sign]

(b + c) + a = (c + b) + a [commutative property of addition--we switched the order of b and c around a plus sign]



The second equation in your question follows from both the commutative and associative properties of multiplication:



a(bc) = a(cb) [commutative property of multiplication--we switched the order of multiplying b and c]

a(cb) = (ac)b [associative property of multiplication--we moved the parentheses]

(ac)b = (ca)b [commutative property of multiplication--we switched the order of multiplying a and c].



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Second question: No. For example:



(4 / 2) / 2 = 2 / 2 = 1; but

4 / (2 / 2) = 4 / 1 = 4.



So (a / b) / c is (most of the time) not equal to a / (b / c); that is, division is not associative, so there is no associative property of division.

Answer 3

The relations you have presented illustrate the commutative property. Just as members of a carpool are still the same carpool even if they sit in different seats, addition yields the same sum if the arguments sit in different seats inside the parentheses and multiplication yields the same product if different arguments sit inside the parentheses.



Associativity is more like three best friends entering a room. One walks ahead of the other two who enter with arms linked. The friends are still a triad when the lone one pairs with one of the others and the third one walks alone.

Answer 4

I would say yes, because all it is is grouping numbers together, and even if you arrange it like that, the value would still be the same. And no, I don't think there's an associative property for division, just like there isn't one for subtraction, because when you divide or subtract with rearranged terms, it's not the same output.

Answer 5

yes. no.