Stupid question on basic inverse operations and properties?

Question

I apologize if my question doesn't make sense. I'm very mathematically illiterate and so I'm not sure how to ask this and have it make sense. Hopefully someone might be able to decipher what I'm saying and help me "see the light" so to speak or better yet point me to the right direction

If subtraction or division are inverse operations then why can't associative and commutative laws be inverse and in some way or weird manner and be applied?

I mean why do inverse operations fail in this way?

If set a(n,n,n) and set b(n,n,n,n) yield the sum of 7 and the order of which they are operated(combined/grouped) doesn't matter "ex: a(n,n,n) + b(n,n,n,n) = b(n,n,n,n) + a(n,n,n) or simply (n,n,n,n,n,n,n) or 7", then why doesn't subtraction which is basically a backwards way of doing addition, allow for some backwards/inverse form of associative or communities properties?

Why does it behave so different?

Answer 1

I'm sorry to say that your example is not quite properly defined. There's no good addition operation that takes a set of 3 elements and adds it to a set of 4 elements in the manner you've described. Most vector addition (in most algebraic rings) would take a(n,n,n) + b(n,n,n,n) to be (a+b)[(2n,2n,2n,n)], which makes the assumption that a can be written a(n,n,n,0). I definitely see what you're doing, but vector addition by tuples is component-based (first coordinate of a added to first coordinate of b, nth coordinate of a added to nth coordinate of b, etc.



Any time you have an additive inverse (a number gained by subtraction from 0 in one way of thinking about it), that number is in the ring itself, and therefore all the same properties (associativity, commutativity, etc.) hold.



a - b is the same as a + (-b), which is equal to (-b) + a. Addition is commutative. a - b is different from b - a because b - a is b + (-a). If you add a-b and b-a, you'll see that they add to zero, so (a - b) and (b - a) are themselves additive inverses. What about this doesn't work? Recall that additive inverses are the same number for 0 only.



The only things guaranteed by these properties are existence (within a ring), and consistency. It would lead to all manner of inconsistencies if (a - b) and (b - a) were the same number, because then they'd both be 0 for all values of a and b, which means that all values of a and b would have to be zero. That isn't a very interesting ring, I'm afraid.



EDIT: Sorry if this is a bit dense, I am coming from a background in abstract algebra. Let me know if there's something I could explain better.

Answer 2

Your question, more concisely, is: addition is associative and subtraction is the inverse of addition, so why isn't subtraction associative? And similarly for multiplcation and division?



A similar question might be: if a+b = b+a how come a-b does not equal b-a? Where now it is commutativity that is at issue.



Another similar question is: It is perfectly OK to multiply by 0, so why is it a problem to divide by 0?



Or this: exponentiation is basically repeated multiplication, multiplication is associative how come exponentiation is not associatve? (For example 2^(3^2) does not equal (2^3)^2 ).



All I can say, is think about the meaning of all these things, and make up examples, and convince yourself that your desire for a perfect world is just not the way things are.

Answer 3

The rigorous mathematical explanation is to to get the exponent degree of x to 1. In order to do that, raise each side to the 1/2, which is the same as taking a square root. Exponent rules tell you to multiply the 2 exponent times the newly included 1/2 exponent to get a product of 1, leaving plain x. Now the other side is reduced to root 16, which is 4 OR -4. Do not forget to include both the positive and negative answer as each is equally as valid.