Question

Show that √7 is an irrational number.in steps

Answer 1

GIVEN:

• √7

TO FIND:

• Prove that √7 is an Irrational number.

SOLUTION:

Suppose √7 represents a Rational number. Then √7 can be expressed in the form of p/q , where p,q are coprimes (q≠0)

$\sf{\longmapsto \sqrt{7} = \dfrac{p}{q} }$

Squaring on both sides

$\sf{\longmapsto 7 = \dfrac{p^2}{q^2}}$

$\sf{\longmapsto 7q^2 = p^2....1) }$

→ 7 divides p²

→ 7 divides p

Let p = 7m

$\sf{\longmapsto p^2 = 49 m^2 }$

Putting the value of p² in 1), we get

$\sf{\longmapsto 7q^2 = 49m^2}$

$\sf{\longmapsto q = 7m^2 }$

→ 7 divides q²

→ 7 divides q

Thus, 7 is common factor of p and q both,

This contradicts our supposition so there is no common factor of p and q.

Hence, √7 is an Irrational number.

______________________

Answer 2

$\huge{\underline{\underline{\red{\bf{Given:}}}}}$

• A irrational number √7 is given to us.

$\rule{200}4$

$\huge{\underline{\underline{\red{\bf{To\: Prove:}}}}}$

• √7 is a Irrational number.

$\rule{200}4$

$\huge{\underline{\underline{\red{\bf{Concept \:Used:}}}}}$

• We will use method of contradiction to prove that √7 is a Irrational number.

$\rule{200}4$

$\huge{\underline{\underline{\red{\bf{Answer:}}}}}$

On the contarary let us assume that √7 is a Rational number .

Then , it can be expressed in the form of p/q where p and q are integers and q≠0 .Also p and q are co - primes i.e. their HCF is 1.

Now as per our assumption,

⇒√7 = p/q.

(Squaring both sides)

⇒(√7)² = (p/q)².

⇒ 7 = p²/q².

⇒7q² = p² . ...........(i)

Now this implies 7 is a factor of p² .Hence from the "Fundamental Theorem of Arithmetic" , 7 will divide p also .

⇒p = 7×k.

⇒p = 7k. ...........(ii)

Putting this value in equation (i) , we have ;

⇒7q² = (7k)².

⇒7q² = 49k².

⇒q² = 49k²/7.

⇒q² = 7k².

This implies 7 is a factor of q² . Hence it will divide q also from the "Fundamental Theorem Arithmetic".

⇒q = 7m. ............(iii)

Hence from above discussion we observe here that 7 is a factor of both p and q . This contradicts our assumption that p and q are co - primes .

Therefore our assumption was wrong √7 is not a Rational number , it is a Irrational number.