Question

Which number can be added to a rational number a to explain that the some of rational number in the in rational number is inrational

Answer 1

You may add, for example,
`√(2)`, or
`\pi`, to any fraction, and the result will be irrational.

In fact, we know that
`√(2)` or
`\pi` can't be written as fraction. So, let's pretend that

`(a)/(b) + √(2) = (c)/(d)`

which means, the sum of a fraction and an irrational is rational. This can't be true, because it would imply

`√(2) = (c)/(d)-(a)/(b) = (bc-ad)/(bd)`

And so we have written
`√(2)` as a fraction, but this is impossible!

So, this proves that the sum of a rational and an irrational is irrational.