Question

prove that 3+2√5 is irrational​

Answer 1

To prove this we need to use the contradiction method.

ABOUT CONTRADICTION :-

In this process we assume something and solve it. We just try to prove that whatever we assumed was wrong.

So, the given situation was wrong and ultimately the actual thing will be converse of it.

So, let's solve!!

TO PROVE :- 3+2root 5 is irrational.

PROOF :- Assume that the given real number is rational. This means that the number can be expressed in the form p/q where p and q belong to integers as well as are co-prime.

So,

3 + 2root5 = p/q

Or,

2root5 = p/q - 3 = (p -3q)/q Or,

Root5 = (p-3q)/2q ....... (i)

Now, (p-3q)/2q is a rational.

So,

irrational number ≠ rational number.

This means root5 is rational.

But, root5 is an irrational.

How??

Assume root5 as rational. So, Root5 = a/b

Where a and b are integers and co-primes.

So,

Squaring both sides:-

5 = p²/q²

So, p² =5q² ...... (ii)

So, p² has 5 as a factor. So, p also has 5 as its factor for some integer c.

Now,

p =5c

Or, p² =25c²

Putting it in (ii)

5q² =25c²

Or, q² = 5c²

So, q² is a multiple of 5 So, q is also a multiple of 5.

Now, Both p and q have a common factor 5 This means they are not co-primes but it is given that they are co-primes.

Hence, it's a contradiction which has risen because of taking root5 as rational.

So, root5 is irrational.

Now,

Back to the question. From (i) :-

Root5 = (p-3q)/2q

So, This is not possible as root5 is irrational and RHS of the equation is rational.

As irrational ≠ rational.

Hence, it is a contradiction.

This has risen because of taking the given number (3 + 2root5) as rational number.

This implies that 3 + 2root5 is an irrational number.