Question

Prove that the sum of a rational number and an irrational number is always irrational.

Answer 1

STEP 1 : Assumption

Let us suppose two numbers $a$ and $b$ which are rational and irrational respectively.
Now, we know that a rational number can be written in the form of fraction ${\displaystyle \frac{p}{q}}$ , where $q\ne 0$. And we know that an irrational number cannot be written in the form of ${\displaystyle \frac{p}{q}}$.

Let us assume that the sum of $a$ and $b$ will give us a rational number. It means we can represent sum of $a$ and $b$ in terms of fraction ${\displaystyle \frac{p}{q}}$, where $q\ne 0$ and $p$ and $q$ 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 ${\displaystyle \frac{p}{q}}$ to $a+b$ and get relation as

$a+b=\frac{p}{q}$

Transpose $a$ to RHS we get

$b=\frac{p}{q}-a$ $...\left(1\right)$

Both ${\displaystyle \frac{p}{q}}$ and $a$ are rational numbers. so $\frac{p}{q}-a$ is a rational number.

According to equation $\left(1\right)$ 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 $b$ 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.