Question

If x= 0 is rational and y irrational, prove that x + y,x − y,xy,x/y and y/x are all irrational.

Answer 1

Here we have $(x\neq0)\in\mathbb{Q}$ and y$\in\mathbb{Q}^c$

• $x+y$

Here let's assume this is rational.

Now, lets take x=3 and y=$\sqrt{2}$

So, $x+y=3+\sqrt{2}$ which is irrational.

So, we have a counter-example to show that $x+y$ is irrational. Hence our claim is false and is irrational.

• $x-y$

Here let's assume this is rational.

Now, lets take x=3 and y=$\sqrt{2}$

So, $x-y=3-\sqrt{2}$ which is irrational.

So, we have a counter-example to show that $x-y$ is irrational. Hence our claim is false and is irrational.

• $xy$

Here let's assume this is rational.

Now, lets take x=3 and y=$\sqrt{2}$

So, $xy=3\sqrt{2}$ which is irrational.

So, we have a counter-example to show that $xy$ is irrational. Hence our claim is false and is irrational.

• $\frac{x}{y}$

Here let's assume this is rational.

Now, lets take x=3 and y=$\sqrt{2}$

So, $\frac{x}{y}=\frac{3}{\sqrt{2}}$ which is irrational.

So, we have a counter-example to show that $\frac{x}{y}$ is irrational. Hence our claim is false and is irrational.

• $\frac{y}{x}$

Here let's assume this is rational.

Now, lets take x=3 and y=$\sqrt{2}$

So, $\frac{y}{x}=\frac{\sqrt{2}}{3}$ which is irrational.

So, we have a counter-example to show that $\frac{y}{x}$ is irrational. Hence our claim is false and is irrational.