Prove that the sum of a rational number and an irrational number is always irrational.
STEP 1 : Assumption
Let us suppose two numbers and which are rational and irrational respectively.
Now, we know that a rational number can be written in the form of fraction , where . And we know that an irrational number cannot be written in the form of .
Let us assume that the sum of and will give us a rational number. It means we can represent sum of and in terms of fraction , where and and are co-prime integers.
STEP 2 : Proving that the sum of a rational number and an irrational number is always irrational
So, we can equate to and get relation as
Transpose to RHS we get
Both and are rational numbers. so is a rational number.
According to equation the left-hand side of the equation is representing a rational number, because the difference between two rational numbers is a rational number.
But we have assumed that is an irrational number.
So, it is not possible that a number is both rational and irrational.
So, it contradicts our assumption that the sum of rational and irrational numbers will be a rational number.
Hence, it is proved that the sum of rational and irrational numbers is an irrational number i.e. opposite to our assumption.