>>11153185what is 1/4 + 1/9?
it is 9/36 + 4/36 = 13/36.
what is 1/4 + 1/9 inverse?
it is 13/36 inverse.
what is 13/36 inverse?
it is 36/13.
is 36/13 the same as 4+9?
no, 36/13 is not the same as 4+9. 4+9 is 169/13.
there, you're wrong.
>then you're using bad misinterpretable notationno, the notation is perfectly clear. you can only apply "inverse" to a number. 1/4 + 1/9 is not a number, it is an expression. but it evaluates to the number 13/36. so when you say 1/4 + 1/9 inverse, we take that to mean 13/36 inverse. as we've exhibited, this is not equal to 1/4 inverse + 1/9 inverse. this is because inversion is not a linear operation.
there certainly are plenty of linear operations. for example, "multiply by 2" is a linear operation. we know from the distributive property that 1/4 + 1/9, multiply by 2, is the same as (1/4 multiplied by 2) + (1/9 multiplied by 2).
but there are plenty of simple nonlinear operations. for instance, inverse is nonlinear. another nonlinear operation might be "add 5". consider this: (1/4 + 1/9) add 5, versus (1/4 add 5) + (1/9 add 5). well suddenly we're adding two 5's instead of just one - these aren't the same thing.
here's a way to see what's going on. what is 1/3? it's 0.33333... which is 3/10 + 3/100 + 3/1000 + 3/10000 + ...
but what is 1/3 inverse? it's 3.
however, 10/3 + 100/3 + 1000/3 + 10000/3 + ... is definitely not 3 - in fact, it's already bigger than 3 when you start with the first element! and it only gets bigger from there. it diverges.
if you want to see a nice reason why 1/1 + 1/4 + 1/9 + 1/16 + ... is pi^2/6 to begin with, i recommend this video:
https://www.youtube.com/watch?v=d-o3eB9sfls 
