Question

Prove that √5-4 is an irrational number.

Answer 1

Step-by-step explanation:

Let us assume that ✓5-4 is a rational number

therefore p/q=✓5-4 (q≠0)

✓5-4=p/q

✓5=p+4/q

As per,p&q p+4/q is integer so,✓5 is also integer

which contradicts that ✓5 is rational number

therefore,our assumption that ✓5-4 is rational is wrong

therefore,✓5-4 is an irrational number

Answer 2

Step-by-step explanation:

On the contrary, let us assume that $\sqrt{5}$ is a rational number. If it is a rational number then we can write it as $\sqrt{5}$ = p/q. p and q are co-primes and q is not equal to 0.

$\sqrt{5}$ = p/q

$\sqrt{5}$q = p

Squaring on both sides

5q^2 = p

q^2 = p^2/5        .............. 1

If 5 divides p^2, it divides p also.      

Let p = 5m

5q^2= 25m^2

q^2 = 5m^2

m^2 = q^2/5          .............. 2

If 5 divides q^2, it divides q also.

From equations 1 and 2, we get that 5 is dividing both p and q.

This contradicts are assumption that p and q are co - primes

Hence $\sqrt{5}$ is irrational.

Now if we subtract 4 from $\sqrt{5}$, it is also irrational as rational number subtracted from irrational number is also irrational.

Hence proved