Thanks Leronz!
This is the first question I have to do:
If a, b, a1, b1, c and d are integers, and if a/b=a1/b1, then a/b + c/d = a1/b1 + c/d and (a/b)*(c/d) = (a1/b1)*(c/d).
It seems very redundant and I always feel like I'm going in circles when trying to do it.
What are your definitions of fractions, fraction addition and multiplication? The point of this particular exercise is most likely to show that the operations are well defined i.e don't depend on which fraction you choose.
is defined as the set of pairs
where
are integers with
not zero. We identify certain pairs, declare
if
, so technically the rationals are a set of equivalence classes of pairs.
We then
define 
but the issue is how do you know that if
and
then is the sum well defined? i.e is the following true:
A priori we don't know that it is i.e when you randomly make up some mathematical object and define some operations, it may be the case that these operations aren't well defined i.e the operations depend on how you represent the elements. So let's check this, according to our definition of equivalence of fractions we must check that the following is
:
b'd' - (a'd'+b'c')bd = adb'd'+bcb'd' - bda'd' - bdb'c' = dd'(ab'-ba') + bb'(cd'-dc') = dd'.0+bb'.0=0)
Where we got the fact that
)
and
)
are 0 is because of the equivalence
and
respectively.
So the point is: Use the definitions of fractions given, only use properties that you know about integers (whole numbers). This is really an exercise in verifying that the construction of fractions is kosher and that they do indeed behave the way we expect from primary school.
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