Question

सिद्ध करे कि रूट 2 एक अपरिमेय संख्या है

Answer 1

Answer:

The proof is explained step-wise below :

Step-by-step explanation:

To prove : √2 is irrational

Proof : We will prove this by using contradiction

Assume √2 is rational that is it can be expressed as a rational fraction of the form :

$\frac{b}{a}$

where a and b are two relatively prime integers.  

$\text{Now, since }\sqrt{2}=\frac{b}{a}\\\\\text{We have }2=\frac{b^2}{a^2}\\\\\implies b^2=2\cdot a^2$

Since 2·a² is even ⇒ b² must be even

And since b² is even ⇒ b is even

Let b = 2·c

⇒ 4·c² = 2·a²

⇒ a² = 2·c²

Since 2·c² is even ⇒ a² is even,

And since a² is even ⇒ a is even

However, two even numbers cannot be relatively prime, so √2 cannot be expressed as a rational fraction. So, we get a contradiction and thus our assumption is wrong.

Hence, √2 is irrational number.

Hence Proved.

Answer 2

Answer:

प्रश्न 1. सिद्ध कीजिए √2 अपरिमेय संख्या है।

उत्तर- यदि सम्भव हो, तो माना √2 एक परिमेय संख्या है।

तब मान √2 = m / n, H.C.F. (m, n) = 1, n≠ 0

⇒ m = √2n

⇒ m2 = 2n2 ….(1)

⇒ 2n2 एक समपूर्णाक है।

⇒ m2 एक समपूर्णांक है।

⇒ m एक समपूर्णांक है। ....(A)

= m = 2q, q∈ z ....(2)

(1) व (2) से

4q2 = 2n2

⇒ n2 = 2q2

⇒ n2 एक समपूर्णांक है।

⇒ n एक समपूर्णांक है। ....(B)

(A) तथा (B) ⇒ m तथा n दोनों ही समपूर्णांक है।

⇒ H.C.F. (m, n) # 1

अतः जो कि विरोधाभास है परिमेय होने का अतः √2 एक अपरिमेय संख्या है।