Advanced Math/Number Sums and their Reciprocals, and why they dont equal out
I am looking for the "why" to this question. I understand that:
a + b = c
1/a + 1/b does not equal 1/c
but if I take 2 numbers and add them together, then take the
inverse of that answer, why will it not be equal to the result,
it I take the inverse of those 2 numbers, add them and take the
inverse of that answer? It just doesnt appear that the math
acts logically. I can do the math and see the result, its the
"why" that puzzles me. If you can explain the "why", I would
really like to know. I hope I explained my puzzle correctly.
5 + 3 = 8
1/8 = 0.125 (answer 1)
1/5 + 1/3 is the same as:
0.2 + 0.3 and = 0.53
and 1/0.53 = 1.88 (answer 2)
answer 1 does not equal answer 2
This is a very fundamental question and it is good you are wondering about it. You feel that the fact that the-sum-of-the-inverses is not equal to the-inverse-of-the-sum is not logical. In this case, the fact that one's faith in algebra results in an seemingly illogical result should lead to questioning the meaning of "logical" as much as perhaps shining a light of suspicion on the laws of algebra. They are not really at odds with each other but, whether they are or not in your mind, depends on the definitions you make and the axioms you assume.
We need to be careful how we define operations, like addition, multiplication and taking the inverse. Assume we have 2 operations * and @ and members of a set S consisting of the elements a1, a2, ... aN. We can loosely think of * as an inverse and @ as addition and the aNs as real numbers for now. In this sense, * is more an an operator on a member (or collection of members) of S and @ is an operation between members of S. Defining your logical statement, we would have
eq. (1) (*a1@*a2) = *(a1@a2)
sum-of-inverses = inverse-of-sum. There is nothing wrong with saying this at this point, but lets see where it takes us. Eq. (1) says that * distributes over @, where the word "distributes" is an honest-to-god math concept that is defined by (1). Lets go ahead and apply * to the left hand side of (1)
eq. (2) *(*a1@*a2) = [*(*a1)]@[*(*a2)] or " 1/(1/a1 +1/a2) = 1/(1/a1) + 1/(1/a2) = a1 + a2 "
where all the brackets and parenthese are used because we haven't yet defined what *(*) means. If we take * to really operate the same as an inverse, then *(*) is the inverse of the inverse and if applied to an element should give back the same element so that
*(*a1) = a1.
Applying this to the left side of (2) gives
and to the right side gives
which says that *a2 = a2, which is OK but puts a pretty big restriction on a2. If we don't want this restriction, it suggests that interpreting * as an inverse and writing down (1) are inconsistent.
The inconsistency of (1) when * and @ are defined as inverse and addition should give you some faith that real algebra really works. This also shows that fully understanding the properties of the operations you define are very important and that the familiarty we have with simple operations can fool us as to their true structure.
With your curious mind, you should study group and ring theory where consistent operations, algebra and properties such as inverses and identity elements generate rich structures that are applicable in the real world. You should also look into Principia Mathematica by Russell and Whitehead in order to cure you of delving too deeply into symbolic logic.
Hope this helped.