Question

Prove that √3 is an irrational number and hence show that 5 - √3 is also an irrational

number.​

Answer 1

Answer:

Let us say that root3 is rational.

Then, root3 = a/b, a and b are rational and co-prime.

b × root3 = a

3b^2 = a^2

So, 3 divides a.

Let us say that a = 3c, c is any rational.

Then, 3b^2 = 9c^2

b^2 = 3c^2

So, 3 divides b.

3 is a common factor of a and b. But a and b are co-primes. This is a contradiction arisen by our assumption that root3 is rational. Hence, root3 is irrational.

Step-by-step explanation:

Let us say that 5 - root3 is rational.

Then, 5 - root3 - 5 will also be rational. [Closure under subtraction]

So, - root3 is rational. So, - root3 × -1 will also be rational. [Closure under Multiplication]

So, root3 is rational.

But this contradicts the fact that root3 is irrational. This has been arisen by our assumption that 5 - root3 is rational. Hence, 5 - root3 is irrational.