Question

prove that 3-√2 is an irrational number​

Answer 1

Let us suppose that 32–√32 is a rational number.

As we know that any rational number can be represented in the form of abab where aa and bb are two co-prime positive integers.

∴32–√=ab⇒2–√=a3b →(1)∴32=ab⇒2=a3b →(1)

Now if we observe the RHS of equation (1) carefully, we can say it is always a rational number because aa and bb are two co-prime positive integers and 3 is also an integer.

Also, the LHS of the equation (1) i.e., 2–√2 is an irrational number.

Therefore, equation (1) is contradicting since LHS is irrational and RHS is rational.

So, our assumption is not correct.

Hence, 32–√32 is irrational.

Note- In these types of problems, the number which needs to be proved irrational is assumed as rational and then equated to the general form of any rational number and if this equation holds true then the given number is rational else irrational.