I am really curious about the Vedanta behind the arithmetic operations on irrational numbers. It still got aggrevated after the productive discussions with my friend. So I decided to ask it here. Basically there is some confusion with some 4 operators.
- Let me start with $+$ operator. There is no confusion. Suppose the "$+$" in $\sqrt{3} +\sqrt{2}$ just add the decimal part of the both numbers. Its like adding $ 1+0.73205... + 1 +0.41421... $ . So it makes sense when we add them linearly .
- The problem arises with the multiplication. Can some one explain how is $\sqrt{3}*\sqrt{3} = 3$ ?. How can a product of two irrational numbers turns out to be a rational number ? . Its like $1.73205.....*1.73205.....$ , so the multiplication operator just multiplies the decimal part too. It must give rise to the infinite decimal part in the output too. But in converse we are getting a rational number ( specifically an integer ) $3$. How can one make sense out of this contradiction ? .
- Similarly coming to Division the same problem arises . But once we have well defined multiplication, division may make some sense because we can always rationalize numerator and denominator. So the problem is just finding the explanation behind the multiplication.
- Similarly how about the exponents ? . Raising an irrational number to the power of another irrational number ? . For example take $\sqrt{3}^{\sqrt{2}}$ . How can one raise the irrational number which has infinite precision to another number which has infinite precision ? .
Thank you. Awaiting for your responses.