Question

Prove that 5 is an irrational, .
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Answer 1

Answer:

Step-by-step explanation:

Let take √5 as rational number

If a and b are two co prime number and b is not equal to 0.

We can write √5 = a/b

Multiply by b both side we get

b√5 = a

To remove root, Squaring on both sides, we get

5b² = a² ……………(1)

Therefore, 5 divides a² and according to theorem of rational number, for any prime number p which is divides a² then it will divide a also.

That means 5 will divide a. So we can write

a = 5c

and plug the value of a in equation (1) we get

5b² = (5c)²

5b² = 25c²

Divide by 25 we get

b²/5 = c²

again using same theorem we get that b will divide by 5

and we have already get that a is divide by 5

but a and b are co prime number. so it is contradicting .

Hence √5 is a non rational number

Answer 2

$\huge\fcolorbox{black}{aqua}{Solution:-}$

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This

contradicts our assumption that they are

co-primes.

Therefore, p/q is not a rational number

√5 is an irrational number.