What will be the sum of irrational numbers?

Answer 1

There is no answer.

If the sum is taken for irrationals positive irrationals only, then the answer is clearly + infinity, since the irrationals increase without limit.

But there are negative irrationals so we need to consider the sum of irrationals from - infinity to + infinity. Each irrational has a matching negative irrational and these two sets are exhaustive. So, each can be paired off, and one might think that the sum of an infinite pairs of zeros is zero, right?

NO, unfortunately not. The logic fails when the sum is over infinitely many terms as is illustrated below:

1-1+1-1 ... = 1+(-1+1)+(-1+1) ... = 1+0+0 ... = 1

or

1-1+1-1 ... = (1-1)+(1-1)+ ... = 0+0+ ... = 0

Incidentally, the above example was used to "prove" that 1 = 0